HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 19.2s

Alternatives: 10

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O 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_i (* cosTheta_O (/ (- -1.0) v))))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

assert(cosTheta_i < cosTheta_O);
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 * (-(-1.0f) / v)))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Fortran

NOTE: cosTheta_i and cosTheta_O 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_i * (costheta_o * (-(-1.0e0) / v)))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(((sinTheta_i * -sinTheta_O) / v)) * (cosTheta_i * (cosTheta_O * (-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. frac-2neg 98.6%

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

   2. div-inv 98.8%

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

   3. distribute-rgt-neg-in 98.8%

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

   4. metadata-eval 98.8%

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

   5. distribute-neg-frac 98.8%

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

   6. metadata-eval 98.8%

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

   7. frac-2neg 98.8%

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

   8. distribute-neg-frac 98.8%

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

   9. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

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

   1. associate-*l* 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i cosTheta_O)
  (*
   (/ (/ 1.0 v) (sinh (/ 1.0 v)))
   (* (exp (* sinTheta_i (/ sinTheta_O v))) (/ 0.5 v)))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * (((1.0f / v) / sinhf((1.0f / v))) * (expf((sinTheta_i * (sinTheta_O / v))) * (0.5f / v)));
}

Fortran

NOTE: cosTheta_i and cosTheta_O 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 * costheta_o) * (((1.0e0 / v) / sinh((1.0e0 / v))) * (exp((sintheta_i * (sintheta_o / v))) * (0.5e0 / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 92.9%

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

   2. exp-neg 92.9%

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

   3. *-commutative 92.9%

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

   4. exp-neg 92.9%

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

   5. *-commutative 92.9%

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

   6. distribute-neg-frac 92.9%

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

   7. distribute-rgt-neg-out 92.9%

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

   8. associate-/l* 92.9%

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

   9. associate-/l* 92.9%

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

   10. *-commutative 92.9%

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

3. Simplified 92.9%

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

   1. div-inv 93.0%

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

   2. times-frac 92.9%

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

   3. clear-num 93.0%

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

5. Applied egg-rr 93.0%

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

   1. associate-/r/ 93.0%

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

   2. *-commutative 93.0%

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

   3. associate-*r/ 93.0%

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

7. Simplified 93.0%

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O 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))) (/ 0.5 v))
  (* cosTheta_i (/ (/ cosTheta_O v) (sinh (/ 1.0 v))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf((sinTheta_i * (sinTheta_O / v))) * (0.5f / v)) * (cosTheta_i * ((cosTheta_O / v) / sinhf((1.0f / v))));
}

Fortran

NOTE: cosTheta_i and cosTheta_O 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))) * (0.5e0 / v)) * (costheta_i * ((costheta_o / v) / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 92.9%

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

   2. exp-neg 92.9%

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

   3. *-commutative 92.9%

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

   4. exp-neg 92.9%

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

   5. *-commutative 92.9%

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

   6. distribute-neg-frac 92.9%

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

   7. distribute-rgt-neg-out 92.9%

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

   8. associate-/l* 92.9%

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

   9. associate-/l* 92.9%

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

   10. *-commutative 92.9%

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

3. Simplified 92.9%

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

   1. div-inv 93.0%

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

   2. times-frac 92.9%

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

   3. clear-num 93.0%

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

5. Applied egg-rr 93.0%

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

   1. associate-/r/ 93.0%

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

   2. *-commutative 93.0%

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

   3. associate-*r/ 93.0%

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

7. Simplified 93.0%

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i
  (*
   (* (exp (* sinTheta_i (/ sinTheta_O v))) (/ 0.5 v))
   (/ (/ cosTheta_O v) (sinh (/ 1.0 v))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i * ((expf((sinTheta_i * (sinTheta_O / v))) * (0.5f / v)) * ((cosTheta_O / v) / sinhf((1.0f / v))));
}

Fortran

NOTE: cosTheta_i and cosTheta_O 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 * ((exp((sintheta_i * (sintheta_o / v))) * (0.5e0 / v)) * ((costheta_o / v) / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 92.9%

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

   2. exp-neg 92.9%

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

   3. *-commutative 92.9%

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

   4. exp-neg 92.9%

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

   5. *-commutative 92.9%

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

   6. distribute-neg-frac 92.9%

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

   7. distribute-rgt-neg-out 92.9%

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

   8. associate-/l* 92.9%

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

   9. associate-/l* 92.9%

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

   10. *-commutative 92.9%

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

3. Simplified 92.9%

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

   1. div-inv 93.0%

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

   2. times-frac 92.9%

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

   3. clear-num 93.0%

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

5. Applied egg-rr 93.0%

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

   1. associate-/r/ 93.0%

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

   2. *-commutative 93.0%

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

   3. associate-*r/ 93.0%

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

7. Simplified 93.0%

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 5: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: cosTheta_i and cosTheta_O 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 / (v ** 2.0e0)) / (sinh((1.0e0 / v)) * (2.0e0 / costheta_o))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.6%

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

   4. associate-*l/ 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. distribute-lft-neg-out 98.6%

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

   7. associate-/l* 98.6%

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

   8. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.6%

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

   2. associate-*l/ 98.6%

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

   3. div-inv 98.8%

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

   4. associate-*l* 98.7%

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

5. Applied egg-rr 98.7%

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

   1. associate-*l* 98.9%

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

   2. associate-*r/ 98.9%

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

7. Simplified 98.9%

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

9. Applied egg-rr 98.3%

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

10. Taylor expanded in sinTheta_i around 0 98.3%

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

11. Final simplification 98.3%

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

Alternative 6: 63.4% accurate, 1.8× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O 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 (/ v (- sinTheta_O))))
  (+
   (* 2.0 (/ v (* cosTheta_i cosTheta_O)))
   (* 0.3333333333333333 (/ 1.0 (* cosTheta_O (* v cosTheta_i)))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((sinTheta_i / (v / -sinTheta_O))) / ((2.0f * (v / (cosTheta_i * cosTheta_O))) + (0.3333333333333333f * (1.0f / (cosTheta_O * (v * cosTheta_i)))));
}

Fortran

NOTE: cosTheta_i and cosTheta_O 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 / (v / -sintheta_o))) / ((2.0e0 * (v / (costheta_i * costheta_o))) + (0.3333333333333333e0 * (1.0e0 / (costheta_o * (v * costheta_i)))))
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(Float32(Float32(2.0) * Float32(v / Float32(cosTheta_i * cosTheta_O))) + Float32(Float32(0.3333333333333333) * Float32(Float32(1.0) / Float32(cosTheta_O * Float32(v * cosTheta_i))))))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((sinTheta_i / (v / -sinTheta_O))) / ((single(2.0) * (v / (cosTheta_i * cosTheta_O))) + (single(0.3333333333333333) * (single(1.0) / (cosTheta_O * (v * 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-/l* 92.9%

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

   2. exp-neg 92.9%

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

   3. *-commutative 92.9%

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

   4. exp-neg 92.9%

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

   5. *-commutative 92.9%

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

   6. distribute-neg-frac 92.9%

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

   7. distribute-rgt-neg-out 92.9%

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

   8. associate-/l* 92.9%

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

   9. associate-/l* 92.9%

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

   10. *-commutative 92.9%

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

3. Simplified 92.9%

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

4. Taylor expanded in v around inf 64.6%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\color{blue}{2 \cdot \frac{v}{cosTheta_O \cdot cosTheta_i} + 0.3333333333333333 \cdot \frac{1}{cosTheta_O \cdot \left(cosTheta_i \cdot v\right)}}} \]

5. Final simplification 64.6%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{2 \cdot \frac{v}{cosTheta_i \cdot cosTheta_O} + 0.3333333333333333 \cdot \frac{1}{cosTheta_O \cdot \left(v \cdot cosTheta_i\right)}} \]

Alternative 7: 59.0% accurate, 24.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{1}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O \cdot 0.5}} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ (/ v cosTheta_i) (* cosTheta_O 0.5))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / ((v / cosTheta_i) / (cosTheta_O * 0.5f));
}

Fortran

NOTE: cosTheta_i and cosTheta_O 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 = 1.0e0 / ((v / costheta_i) / (costheta_o * 0.5e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.6%

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

   4. associate-*l/ 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. distribute-lft-neg-out 98.6%

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

   7. associate-/l* 98.6%

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

   8. *-commutative 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 59.5%

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

   1. associate-*r/ 59.5%

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

6. Simplified 59.5%

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

7. Taylor expanded in cosTheta_O around 0 59.5%

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

   1. associate-/l* 59.5%

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

9. Simplified 59.5%

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

    1. associate-*r/ 59.4%

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

    2. clear-num 60.1%

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

11. Applied egg-rr 60.1%

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

12. Final simplification 60.1%

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

Alternative 8: 58.5% accurate, 31.4× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O 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_O (/ cosTheta_i v))))

C

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

Fortran

NOTE: cosTheta_i and cosTheta_O 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_o * (costheta_i / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.6%

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

   4. associate-*l/ 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. distribute-lft-neg-out 98.6%

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

   7. associate-/l* 98.6%

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

   8. *-commutative 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 59.5%

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

   1. associate-*r/ 59.5%

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

6. Simplified 59.5%

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

7. Final simplification 59.5%

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

Alternative 9: 58.5% accurate, 31.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ 0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v} \end{array} \]

FPCore

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.6%

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

   4. associate-*l/ 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. distribute-lft-neg-out 98.6%

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

   7. associate-/l* 98.6%

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

   8. *-commutative 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 59.5%

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

5. Final simplification 59.5%

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

Alternative 10: 59.0% accurate, 31.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \end{array} \]

FPCore

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

C

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

Fortran

NOTE: cosTheta_i and cosTheta_O 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 / (v / (costheta_i * costheta_o))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.6%

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

   4. associate-*l/ 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. distribute-lft-neg-out 98.6%

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

   7. associate-/l* 98.6%

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

   8. *-commutative 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 59.5%

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

   1. associate-*r/ 59.5%

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

6. Simplified 59.5%

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

   1. *-commutative 59.5%

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

   2. associate-/r/ 59.5%

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

   3. clear-num 60.1%

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

   4. associate-*r/ 60.1%

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

   5. metadata-eval 60.1%

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

   6. associate-/l/ 60.1%

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

   7. *-commutative 60.1%

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

8. Applied egg-rr 60.1%

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

9. Final simplification 60.1%

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

Reproduce

herbie shell --seed 2023301 
(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)))