HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.9%

Time: 17.5s

Alternatives: 12

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: 98.9% 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 98.5%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

   1. div-inv 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.6%

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

   1. div-inv 98.7%

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

8. Applied egg-rr 98.7%

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

9. Final simplification 98.7%

   \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(\frac{1}{v} \cdot \frac{1}{v}\right)\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
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 98.5%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

   1. div-inv 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

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_O (- sinTheta_i)) 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_O * -sinTheta_i) / 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_o * -sintheta_i) / 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_O * Float32(-sinTheta_i)) / 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_O * -sinTheta_i) / v)) * ((single(1.0) / v) * (cosTheta_O * cosTheta_i))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. div-inv 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

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_O (- sinTheta_i)) v)) (* cosTheta_O (/ cosTheta_i 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 (expf(((sinTheta_O * -sinTheta_i) / v)) * (cosTheta_O * (cosTheta_i / 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_o * -sintheta_i) / v)) * (costheta_o * (costheta_i / 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(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(cosTheta_O * Float32(cosTheta_i / v))) / 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_O * -sinTheta_i) / v)) * (cosTheta_O * (cosTheta_i / v))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. div-inv 98.6%

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in cosTheta_i around 0 98.5%

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

   1. associate-*r/ 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

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_O (- sinTheta_i)) v)) (/ (* cosTheta_O cosTheta_i) 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 (expf(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_O * cosTheta_i) / 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_o * -sintheta_i) / v)) * ((costheta_o * costheta_i) / 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(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(cosTheta_O * cosTheta_i) / v)) / 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_O * -sinTheta_i) / v)) * ((cosTheta_O * cosTheta_i) / v)) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.5%

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

3. Final simplification 98.5%

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

Alternative 6: 64.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{cosTheta\_O}{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \end{array} \]

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 (exp (* sinTheta_O (/ sinTheta_i v))))
  (/ cosTheta_i (fma v 2.0 (/ 0.3333333333333333 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 / expf((sinTheta_O * (sinTheta_i / v)))) * (cosTheta_i / fmaf(v, 2.0f, (0.3333333333333333f / v)));
}

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 / exp(Float32(sinTheta_O * Float32(sinTheta_i / v)))) * Float32(cosTheta_i / fma(v, Float32(2.0), Float32(Float32(0.3333333333333333) / v))))
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 60.7%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Taylor expanded in cosTheta_i around 0 60.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}} \cdot \left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)}} \]
7. Step-by-step derivation

   1. times-frac 60.7%

      \[\leadsto \color{blue}{\frac{cosTheta\_O}{e^{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \cdot \frac{cosTheta\_i}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]

   2. associate-/l* 60.7%

      \[\leadsto \frac{cosTheta\_O}{e^{\color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}}} \cdot \frac{cosTheta\_i}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}} \]

   3. associate-*r/ 60.7%

      \[\leadsto \frac{cosTheta\_O}{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \cdot \frac{cosTheta\_i}{2 \cdot v + \color{blue}{\frac{0.3333333333333333 \cdot 1}{v}}} \]

   4. metadata-eval 60.7%

      \[\leadsto \frac{cosTheta\_O}{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \cdot \frac{cosTheta\_i}{2 \cdot v + \frac{\color{blue}{0.3333333333333333}}{v}} \]

   5. *-commutative 60.7%

      \[\leadsto \frac{cosTheta\_O}{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \cdot \frac{cosTheta\_i}{\color{blue}{v \cdot 2} + \frac{0.3333333333333333}{v}} \]

   6. fma-undefine 60.7%

      \[\leadsto \frac{cosTheta\_O}{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \cdot \frac{cosTheta\_i}{\color{blue}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}} \]

8. Simplified 60.7%

   \[\leadsto \color{blue}{\frac{cosTheta\_O}{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}} \]

9. Final simplification 60.7%

   \[\leadsto \frac{cosTheta\_O}{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}} \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \]
10. Add Preprocessing

Alternative 7: 64.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{-1}{v} - v \cdot 2\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot 2 + \frac{1}{v} \cdot 0.3333333333333333} \end{array} \]

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 (* sinTheta_i sinTheta_O)))
   (* v (- (* 0.3333333333333333 (/ -1.0 v)) (* v 2.0))))
  (/
   (* cosTheta_O cosTheta_i)
   (+ (* v 2.0) (* (/ 1.0 v) 0.3333333333333333)))))

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 * (sinTheta_i * sinTheta_O))) / (v * ((0.3333333333333333f * (-1.0f / v)) - (v * 2.0f)))) + ((cosTheta_O * cosTheta_i) / ((v * 2.0f) + ((1.0f / v) * 0.3333333333333333f)));
}

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 * (sintheta_i * sintheta_o))) / (v * ((0.3333333333333333e0 * ((-1.0e0) / v)) - (v * 2.0e0)))) + ((costheta_o * costheta_i) / ((v * 2.0e0) + ((1.0e0 / v) * 0.3333333333333333e0)))
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(Float32(cosTheta_O * Float32(cosTheta_i * Float32(sinTheta_i * sinTheta_O))) / Float32(v * Float32(Float32(Float32(0.3333333333333333) * Float32(Float32(-1.0) / v)) - Float32(v * Float32(2.0))))) + Float32(Float32(cosTheta_O * cosTheta_i) / Float32(Float32(v * Float32(2.0)) + Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333)))))
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 * (sinTheta_i * sinTheta_O))) / (v * ((single(0.3333333333333333) * (single(-1.0) / v)) - (v * single(2.0))))) + ((cosTheta_O * cosTheta_i) / ((v * single(2.0)) + ((single(1.0) / v) * single(0.3333333333333333))));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 60.7%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Taylor expanded in sinTheta_i around 0 60.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v \cdot \left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]

7. Final simplification 60.7%

   \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{-1}{v} - v \cdot 2\right)} + \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot 2 + \frac{1}{v} \cdot 0.3333333333333333} \]
8. Add Preprocessing

Alternative 8: 64.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := \frac{1}{v} \cdot 0.3333333333333333\\ \frac{cosTheta\_O \cdot cosTheta\_i}{t\_0 + \left(v \cdot 2 + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(v \cdot 2 + t\_0\right)\right)}{v}\right)} \end{array} \end{array} \]

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
 (let* ((t_0 (* (/ 1.0 v) 0.3333333333333333)))
   (/
    (* cosTheta_O cosTheta_i)
    (+
     t_0
     (+ (* v 2.0) (/ (* sinTheta_O (* sinTheta_i (+ (* v 2.0) t_0))) 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) {
	float t_0 = (1.0f / v) * 0.3333333333333333f;
	return (cosTheta_O * cosTheta_i) / (t_0 + ((v * 2.0f) + ((sinTheta_O * (sinTheta_i * ((v * 2.0f) + t_0))) / 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
    real(4) :: t_0
    t_0 = (1.0e0 / v) * 0.3333333333333333e0
    code = (costheta_o * costheta_i) / (t_0 + ((v * 2.0e0) + ((sintheta_o * (sintheta_i * ((v * 2.0e0) + t_0))) / 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)
	t_0 = Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333))
	return Float32(Float32(cosTheta_O * cosTheta_i) / Float32(t_0 + Float32(Float32(v * Float32(2.0)) + Float32(Float32(sinTheta_O * Float32(sinTheta_i * Float32(Float32(v * Float32(2.0)) + t_0))) / 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)
	t_0 = (single(1.0) / v) * single(0.3333333333333333);
	tmp = (cosTheta_O * cosTheta_i) / (t_0 + ((v * single(2.0)) + ((sinTheta_O * (sinTheta_i * ((v * single(2.0)) + t_0))) / v)));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 60.7%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Taylor expanded in sinTheta_i around 0 60.7%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{0.3333333333333333 \cdot \frac{1}{v} + \left(2 \cdot v + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)\right)}{v}\right)}} \]

7. Final simplification 60.7%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\frac{1}{v} \cdot 0.3333333333333333 + \left(v \cdot 2 + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(v \cdot 2 + \frac{1}{v} \cdot 0.3333333333333333\right)\right)}{v}\right)} \]
8. Add Preprocessing

Alternative 9: 64.1% accurate, 16.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot 2 + \frac{1}{v} \cdot 0.3333333333333333} \end{array} \]

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 2.0) (* (/ 1.0 v) 0.3333333333333333))))

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 * 2.0f) + ((1.0f / v) * 0.3333333333333333f));
}

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 * 2.0e0) + ((1.0e0 / v) * 0.3333333333333333e0))
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 * cosTheta_i) / Float32(Float32(v * Float32(2.0)) + Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333))))
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(2.0)) + ((single(1.0) / v) * single(0.3333333333333333)));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 60.7%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Taylor expanded in sinTheta_i around 0 60.7%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]

7. Final simplification 60.7%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot 2 + \frac{1}{v} \cdot 0.3333333333333333} \]
8. Add Preprocessing

Alternative 10: 58.4% accurate, 20.0× speedup

Math

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

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) (* 2.0 (+ v (* sinTheta_i sinTheta_O)))))

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) / (2.0f * (v + (sinTheta_i * sinTheta_O)));
}

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) / (2.0e0 * (v + (sintheta_i * sintheta_o)))
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 * cosTheta_i) / Float32(Float32(2.0) * Float32(v + Float32(sinTheta_i * sinTheta_O))))
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(2.0) * (v + (sinTheta_i * sinTheta_O)));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 60.7%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}\right)} \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \]

6. Taylor expanded in v around inf 54.8%

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

   1. +-commutative 54.8%

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

   2. distribute-lft-out 54.8%

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

8. Simplified 54.8%

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

9. Final simplification 54.8%

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

Alternative 11: 58.4% accurate, 31.4× speedup

Math

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

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(Float32(cosTheta_O * 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 98.5%

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

3. Taylor expanded in v around inf 54.7%

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

   1. div-inv 54.7%

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

   2. exp-neg 54.7%

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

   3. associate-/l* 54.7%

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

   4. pow-exp 54.7%

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

   5. pow-flip 54.7%

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

   6. associate-*r/ 54.7%

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

   7. metadata-eval 54.7%

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

5. Applied egg-rr 54.7%

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

   1. *-commutative 54.7%

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

   2. associate-*r* 54.7%

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

   3. exp-prod 54.7%

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

   4. distribute-neg-frac2 54.7%

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

   5. associate-/l* 54.7%

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

   6. *-commutative 54.7%

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

   7. distribute-neg-frac2 54.7%

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

   8. *-commutative 54.7%

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

   9. distribute-neg-frac2 54.7%

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

   10. *-commutative 54.7%

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

   11. distribute-frac-neg2 54.7%

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

   12. *-commutative 54.7%

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

   13. associate-/l* 54.7%

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

   14. distribute-rgt-neg-in 54.7%

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

7. Simplified 54.7%

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

8. Taylor expanded in sinTheta_O around 0 54.7%

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

   1. associate-/l* 54.7%

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

10. Simplified 54.7%

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

11. Final simplification 54.7%

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

Alternative 12: 58.4% accurate, 31.4× speedup

Math

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

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
 (* 0.5 (* cosTheta_i (/ cosTheta_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 0.5f * (cosTheta_i * (cosTheta_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 = 0.5e0 * (costheta_i * (costheta_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(0.5) * Float32(cosTheta_i * Float32(cosTheta_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 = single(0.5) * (cosTheta_i * (cosTheta_O / v));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf 54.7%

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

   1. div-inv 54.7%

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

   2. exp-neg 54.7%

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

   3. associate-/l* 54.7%

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

   4. pow-exp 54.7%

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

   5. pow-flip 54.7%

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

   6. associate-*r/ 54.7%

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

   7. metadata-eval 54.7%

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

5. Applied egg-rr 54.7%

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

   1. *-commutative 54.7%

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

   2. associate-*r* 54.7%

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

   3. exp-prod 54.7%

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

   4. distribute-neg-frac2 54.7%

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

   5. associate-/l* 54.7%

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

   6. *-commutative 54.7%

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

   7. distribute-neg-frac2 54.7%

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

   8. *-commutative 54.7%

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

   9. distribute-neg-frac2 54.7%

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

   10. *-commutative 54.7%

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

   11. distribute-frac-neg2 54.7%

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

   12. *-commutative 54.7%

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

   13. associate-/l* 54.7%

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

   14. distribute-rgt-neg-in 54.7%

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

7. Simplified 54.7%

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

8. Taylor expanded in sinTheta_O around 0 54.7%

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

9. Final simplification 54.7%

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

Reproduce

herbie shell --seed 2024041 
(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)))