HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 17.9s

Alternatives: 12

Speedup: 1.9×

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: 98.7% accurate, 1.0× speedup

Math

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

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) (/ v sinTheta_O))) (* (sinh (/ 1.0 v)) 2.0))
  (* (/ 1.0 v) (* (/ cosTheta_i v) 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 (expf((-sinTheta_i / (v / sinTheta_O))) / (sinhf((1.0f / v)) * 2.0f)) * ((1.0f / v) * ((cosTheta_i / v) * 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 = (exp((-sintheta_i / (v / sintheta_o))) / (sinh((1.0e0 / v)) * 2.0e0)) * ((1.0e0 / v) * ((costheta_i / v) * 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(exp(Float32(Float32(-sinTheta_i) / Float32(v / sinTheta_O))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))) * Float32(Float32(Float32(1.0) / v) * Float32(Float32(cosTheta_i / v) * 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 = (exp((-sinTheta_i / (v / sinTheta_O))) / (sinh((single(1.0) / v)) * single(2.0))) * ((single(1.0) / v) * ((cosTheta_i / v) * cosTheta_O));
end

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. *-commutative 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.8%

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

   2. associate-/r/ 98.8%

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

5. Applied egg-rr 98.8%

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

6. Final simplification 98.8%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

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
 (*
  (* (/ 1.0 v) (* (/ cosTheta_i v) cosTheta_O))
  (/ 1.0 (- (exp (/ 1.0 v)) (exp (/ -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 ((1.0f / v) * ((cosTheta_i / v) * cosTheta_O)) * (1.0f / (expf((1.0f / v)) - expf((-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 = ((1.0e0 / v) * ((costheta_i / v) * costheta_o)) * (1.0e0 / (exp((1.0e0 / v)) - exp(((-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(Float32(Float32(1.0) / v) * Float32(Float32(cosTheta_i / v) * cosTheta_O)) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) - exp(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 = ((single(1.0) / v) * ((cosTheta_i / v) * cosTheta_O)) * (single(1.0) / (exp((single(1.0) / v)) - exp((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. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. *-commutative 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.8%

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

   2. associate-/r/ 98.8%

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

5. Applied egg-rr 98.8%

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

6. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

   3. metadata-eval 98.6%

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

8. Simplified 98.6%

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

9. Final simplification 98.6%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.6%

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

   2. associate-*l/ 98.6%

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

   3. associate-/l/ 98.6%

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

   4. associate-*r/ 98.6%

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

   5. *-commutative 98.6%

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

   6. /-rgt-identity 98.6%

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

   7. associate-/r/ 98.6%

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

   8. exp-neg 98.6%

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

   9. remove-double-div 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

   12. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_i around 0 98.6%

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

   1. times-frac 98.7%

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

   2. unpow2 98.7%

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

   3. associate-*r/ 98.7%

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

6. Simplified 98.7%

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

7. Final simplification 98.7%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.6%

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

   2. associate-*l/ 98.6%

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

   3. associate-/l/ 98.6%

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

   4. associate-*r/ 98.6%

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

   5. *-commutative 98.6%

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

   6. /-rgt-identity 98.6%

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

   7. associate-/r/ 98.6%

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

   8. exp-neg 98.6%

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

   9. remove-double-div 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

   12. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_i around 0 98.6%

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

   1. times-frac 98.7%

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

   2. unpow2 98.7%

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

   3. associate-*r/ 98.7%

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

6. Simplified 98.7%

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

7. Taylor expanded in cosTheta_i around 0 98.7%

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

   1. unpow2 98.7%

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

   2. associate-/r* 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 5: 98.6% accurate, 1.9× speedup

Math

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

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 (/ v cosTheta_O)) (/ (/ 1.0 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 (cosTheta_i / (v / cosTheta_O)) * ((1.0f / 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 = (costheta_i / (v / costheta_o)) * ((1.0e0 / 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(cosTheta_i / Float32(v / cosTheta_O)) * Float32(Float32(Float32(1.0) / 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 = (cosTheta_i / (v / cosTheta_O)) * ((single(1.0) / v) / (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.2%

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

4. Taylor expanded in v around inf 97.9%

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

   1. associate-/l/ 98.4%

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

   2. div-inv 98.6%

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

   3. times-frac 98.6%

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

   4. associate-/l* 98.6%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 6: 98.5% accurate, 1.9× speedup

Math

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

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 v) (/ (/ cosTheta_O 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 (cosTheta_i / v) * ((cosTheta_O / 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 = (costheta_i / v) * ((costheta_o / 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(cosTheta_i / v) * Float32(Float32(cosTheta_O / 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 = (cosTheta_i / v) * ((cosTheta_O / v) / (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.2%

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

4. Taylor expanded in v around inf 97.9%

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

   1. associate-/l/ 98.4%

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

   2. div-inv 98.6%

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

   3. associate-*l* 98.5%

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

   4. times-frac 98.6%

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

   5. un-div-inv 98.5%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

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
 (* (/ cosTheta_O v) (/ (/ cosTheta_i (* (sinh (/ 1.0 v)) 2.0)) 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_O / v) * ((cosTheta_i / (sinhf((1.0f / v)) * 2.0f)) / 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_o / v) * ((costheta_i / (sinh((1.0e0 / v)) * 2.0e0)) / 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(cosTheta_O / v) * Float32(Float32(cosTheta_i / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))) / 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_O / v) * ((cosTheta_i / (sinh((single(1.0) / v)) * single(2.0))) / v);
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.2%

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

4. Taylor expanded in v around inf 97.9%

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

   1. associate-/l/ 97.9%

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

   2. times-frac 98.1%

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

6. Applied egg-rr 98.1%

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

   1. *-un-lft-identity 98.1%

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

   2. associate-/l* 98.5%

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

8. Applied egg-rr 98.5%

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

   1. *-lft-identity 98.5%

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

   2. associate-/r/ 98.5%

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

   3. *-commutative 98.5%

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

10. Simplified 98.5%

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

11. Final simplification 98.5%

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

Alternative 8: 98.5% accurate, 1.9× speedup

Math

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

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
 (/ (/ cosTheta_i (* (sinh (/ 1.0 v)) 2.0)) (/ v (/ 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 (cosTheta_i / (sinhf((1.0f / v)) * 2.0f)) / (v / (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 = (costheta_i / (sinh((1.0e0 / v)) * 2.0e0)) / (v / (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(cosTheta_i / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))) / Float32(v / Float32(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 = (cosTheta_i / (sinh((single(1.0) / v)) * single(2.0))) / (v / (cosTheta_O / v));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.2%

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

4. Taylor expanded in v around inf 97.9%

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

   1. associate-/l/ 97.9%

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

   2. times-frac 98.1%

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

6. Applied egg-rr 98.1%

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

   1. *-un-lft-identity 98.1%

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

   2. associate-/l* 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 9: 58.7% accurate, 31.4× speedup

Math

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

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_i v) 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 * ((cosTheta_i / v) * 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 * ((costheta_i / v) * 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(Float32(cosTheta_i / v) * 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) * ((cosTheta_i / v) * cosTheta_O);
end

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. *-commutative 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 62.1%

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

   1. associate-*l/ 62.1%

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

   2. *-commutative 62.1%

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

6. Simplified 62.1%

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

7. Final simplification 62.1%

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

Alternative 10: 58.7% accurate, 31.4× speedup

Math

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

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.2%

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

4. Taylor expanded in v around inf 97.9%

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

5. Taylor expanded in v around inf 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/l* 62.1%

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

7. Simplified 62.1%

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

8. Final simplification 62.1%

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

Alternative 11: 58.8% 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. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. *-commutative 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 62.1%

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

5. Final simplification 62.1%

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

Alternative 12: 59.1% 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. times-frac 98.6%

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

   2. exp-neg 98.6%

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

   3. *-commutative 98.6%

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

   4. exp-neg 98.6%

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

   5. distribute-neg-frac 98.6%

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

   6. *-commutative 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-/l* 98.6%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 62.1%

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

   1. clear-num 63.0%

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

   2. un-div-inv 63.0%

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

6. Applied egg-rr 63.0%

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

7. Final simplification 63.0%

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

Reproduce

herbie shell --seed 2023264 
(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)))