HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.7%

Time: 11.8s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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_o * -sintheta_i) / v)) * ((costheta_i * costheta_o) * (1.0e0 / v))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. div-inv 98.6%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) / ((sinhf((1.0f / v)) * (v * (v * 2.0f))) * expf((sinTheta_i * (sinTheta_O / 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 = (costheta_i * costheta_o) / ((sinh((1.0e0 / v)) * (v * (v * 2.0e0))) * exp((sintheta_i * (sintheta_o / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. associate-*l/ 98.4%

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

   2. times-frac 98.5%

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

   3. exp-neg 98.5%

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

   4. associate-*l/ 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate-/l/ 98.5%

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

   7. associate-*l* 98.5%

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

   8. associate-*l* 98.5%

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

   9. *-commutative 98.5%

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

   10. *-commutative 98.5%

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

   11. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i / (v * v)) * (cosTheta_O / (expf((1.0f / v)) - expf((-1.0f / 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 = (costheta_i / (v * v)) * (costheta_o / (exp((1.0e0 / v)) - exp(((-1.0e0) / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. associate-*l/ 98.4%

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

   2. times-frac 98.5%

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

   3. exp-neg 98.5%

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

   4. associate-*l/ 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate-/l/ 98.5%

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

   7. associate-*l* 98.5%

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

   8. associate-*l* 98.5%

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

   9. *-commutative 98.5%

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

   10. *-commutative 98.5%

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

   11. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{{v}^{2}} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

   2. unpow2 98.4%

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

   3. rec-exp 98.4%

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

   4. distribute-neg-frac 98.4%

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

   5. metadata-eval 98.4%

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

6. Simplified 98.4%

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

7. Final simplification 98.4%

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

Alternative 4: 71.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \cdot \left(\left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{1}{v}\right)}{2 + \frac{0.3333333333333333}{v \cdot v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   (/ (* (/ cosTheta_O v) (/ cosTheta_i v)) (+ (exp (/ 1.0 v)) -1.0))
   (/
    (*
     (exp (/ (* sinTheta_O (- sinTheta_i)) v))
     (* (* cosTheta_i cosTheta_O) (/ 1.0 v)))
    (+ 2.0 (/ 0.3333333333333333 (* v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = ((cosTheta_O / v) * (cosTheta_i / v)) / (expf((1.0f / v)) + -1.0f);
	} else {
		tmp = (expf(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_i * cosTheta_O) * (1.0f / v))) / (2.0f + (0.3333333333333333f / (v * v)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = ((costheta_o / v) * (costheta_i / v)) / (exp((1.0e0 / v)) + (-1.0e0))
    else
        tmp = (exp(((sintheta_o * -sintheta_i) / v)) * ((costheta_i * costheta_o) * (1.0e0 / v))) / (2.0e0 + (0.3333333333333333e0 / (v * v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(Float32(Float32(cosTheta_O / v) * Float32(cosTheta_i / v)) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)));
	else
		tmp = Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / v))) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = ((cosTheta_O / v) * (cosTheta_i / v)) / (exp((single(1.0) / v)) + single(-1.0));
	else
		tmp = (exp(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_i * cosTheta_O) * (single(1.0) / v))) / (single(2.0) + (single(0.3333333333333333) / (v * v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   1. Initial program 98.1%

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

      1. associate-*l/ 98.1%

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

      2. times-frac 98.0%

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

      3. exp-neg 98.1%

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

      4. associate-*l/ 98.1%

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

      5. *-lft-identity 98.1%

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

      6. associate-/l/ 98.1%

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

      7. associate-*l* 98.1%

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

      8. associate-*l* 98.1%

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

      9. *-commutative 98.1%

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

      10. *-commutative 98.1%

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

      11. associate-*l/ 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in sinTheta_O around 0 97.9%

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

      1. associate-/r* 97.9%

         \[\leadsto \color{blue}{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{v}^{2}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

      2. *-commutative 97.9%

         \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{{v}^{2}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

      3. unpow2 97.9%

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

      4. times-frac 98.0%

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

      5. rec-exp 98.0%

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

      6. distribute-neg-frac 98.0%

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

      7. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in v around inf 73.9%

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

   if 0.449999988 < v

   1. Initial program 98.9%

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

      1. div-inv 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in v around inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. metadata-eval 69.2%

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

      3. unpow2 69.2%

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

   6. Simplified 69.2%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(\left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{1}{v}\right)}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \cdot \left(\left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{1}{v}\right)}{2 + \frac{0.3333333333333333}{v \cdot v}}\\ \end{array} \]

Alternative 5: 71.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.44999998807907104)
   (/ (* (/ cosTheta_O v) (/ cosTheta_i v)) (+ (exp (/ 1.0 v)) -1.0))
   (/
    (*
     (exp (/ (* sinTheta_O (- sinTheta_i)) v))
     (/ (* cosTheta_i cosTheta_O) v))
    (+ 2.0 (/ 0.3333333333333333 (* v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = ((cosTheta_O / v) * (cosTheta_i / v)) / (expf((1.0f / v)) + -1.0f);
	} else {
		tmp = (expf(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_i * cosTheta_O) / v)) / (2.0f + (0.3333333333333333f / (v * v)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (v <= 0.44999998807907104e0) then
        tmp = ((costheta_o / v) * (costheta_i / v)) / (exp((1.0e0 / v)) + (-1.0e0))
    else
        tmp = (exp(((sintheta_o * -sintheta_i) / v)) * ((costheta_i * costheta_o) / v)) / (2.0e0 + (0.3333333333333333e0 / (v * v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(Float32(Float32(cosTheta_O / v) * Float32(cosTheta_i / v)) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)));
	else
		tmp = Float32(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = ((cosTheta_O / v) * (cosTheta_i / v)) / (exp((single(1.0) / v)) + single(-1.0));
	else
		tmp = (exp(((sinTheta_O * -sinTheta_i) / v)) * ((cosTheta_i * cosTheta_O) / v)) / (single(2.0) + (single(0.3333333333333333) / (v * v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   1. Initial program 98.1%

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

      1. associate-*l/ 98.1%

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

      2. times-frac 98.0%

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

      3. exp-neg 98.1%

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

      4. associate-*l/ 98.1%

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

      5. *-lft-identity 98.1%

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

      6. associate-/l/ 98.1%

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

      7. associate-*l* 98.1%

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

      8. associate-*l* 98.1%

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

      9. *-commutative 98.1%

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

      10. *-commutative 98.1%

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

      11. associate-*l/ 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in sinTheta_O around 0 97.9%

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

      1. associate-/r* 97.9%

         \[\leadsto \color{blue}{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{v}^{2}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

      2. *-commutative 97.9%

         \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{{v}^{2}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

      3. unpow2 97.9%

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

      4. times-frac 98.0%

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

      5. rec-exp 98.0%

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

      6. distribute-neg-frac 98.0%

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

      7. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in v around inf 73.9%

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

   if 0.449999988 < v

   1. Initial program 98.9%

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

   2. Taylor expanded in v around inf 69.2%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 69.2%

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

      2. metadata-eval 69.2%

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

      3. unpow2 69.2%

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

   4. Simplified 69.2%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}}\\ \end{array} \]

Alternative 6: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \frac{0.3333333333333333}{v \cdot v}\\ \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{t_0} - \frac{sinTheta_i}{v \cdot v} \cdot \frac{cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 2.0 (/ 0.3333333333333333 (* v v)))))
   (if (<= v 0.44999998807907104)
     (/ (* (/ cosTheta_O v) (/ cosTheta_i v)) (+ (exp (/ 1.0 v)) -1.0))
     (-
      (* (/ cosTheta_i v) (/ cosTheta_O t_0))
      (*
       (/ sinTheta_i (* v v))
       (/ (* cosTheta_i (* sinTheta_O cosTheta_O)) t_0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 2.0f + (0.3333333333333333f / (v * v));
	float tmp;
	if (v <= 0.44999998807907104f) {
		tmp = ((cosTheta_O / v) * (cosTheta_i / v)) / (expf((1.0f / v)) + -1.0f);
	} else {
		tmp = ((cosTheta_i / v) * (cosTheta_O / t_0)) - ((sinTheta_i / (v * v)) * ((cosTheta_i * (sinTheta_O * cosTheta_O)) / t_0));
	}
	return tmp;
}

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = 2.0e0 + (0.3333333333333333e0 / (v * v))
    if (v <= 0.44999998807907104e0) then
        tmp = ((costheta_o / v) * (costheta_i / v)) / (exp((1.0e0 / v)) + (-1.0e0))
    else
        tmp = ((costheta_i / v) * (costheta_o / t_0)) - ((sintheta_i / (v * v)) * ((costheta_i * (sintheta_o * costheta_o)) / t_0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))
	tmp = Float32(0.0)
	if (v <= Float32(0.44999998807907104))
		tmp = Float32(Float32(Float32(cosTheta_O / v) * Float32(cosTheta_i / v)) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)));
	else
		tmp = Float32(Float32(Float32(cosTheta_i / v) * Float32(cosTheta_O / t_0)) - Float32(Float32(sinTheta_i / Float32(v * v)) * Float32(Float32(cosTheta_i * Float32(sinTheta_O * cosTheta_O)) / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(2.0) + (single(0.3333333333333333) / (v * v));
	tmp = single(0.0);
	if (v <= single(0.44999998807907104))
		tmp = ((cosTheta_O / v) * (cosTheta_i / v)) / (exp((single(1.0) / v)) + single(-1.0));
	else
		tmp = ((cosTheta_i / v) * (cosTheta_O / t_0)) - ((sinTheta_i / (v * v)) * ((cosTheta_i * (sinTheta_O * cosTheta_O)) / t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.449999988

   1. Initial program 98.1%

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

      1. associate-*l/ 98.1%

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

      2. times-frac 98.0%

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

      3. exp-neg 98.1%

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

      4. associate-*l/ 98.1%

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

      5. *-lft-identity 98.1%

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

      6. associate-/l/ 98.1%

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

      7. associate-*l* 98.1%

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

      8. associate-*l* 98.1%

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

      9. *-commutative 98.1%

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

      10. *-commutative 98.1%

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

      11. associate-*l/ 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in sinTheta_O around 0 97.9%

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

      1. associate-/r* 97.9%

         \[\leadsto \color{blue}{\frac{\frac{cosTheta_i \cdot cosTheta_O}{{v}^{2}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

      2. *-commutative 97.9%

         \[\leadsto \frac{\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{{v}^{2}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

      3. unpow2 97.9%

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

      4. times-frac 98.0%

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

      5. rec-exp 98.0%

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

      6. distribute-neg-frac 98.0%

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

      7. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in v around inf 73.9%

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

   if 0.449999988 < v

   1. Initial program 98.9%

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

   2. Taylor expanded in v around inf 69.2%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 69.2%

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

      2. metadata-eval 69.2%

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

      3. unpow2 69.2%

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

   4. Simplified 69.2%

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

   5. Taylor expanded in sinTheta_i around 0 69.1%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{{v}^{2} \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 69.1%

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

      2. unsub-neg 69.1%

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

      3. times-frac 69.2%

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

      4. unpow2 69.2%

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

      5. associate-*r/ 69.2%

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

      6. metadata-eval 69.2%

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

      7. unpow2 69.2%

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

      8. times-frac 69.2%

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

   7. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + \frac{0.3333333333333333}{v \cdot v}} - \frac{sinTheta_i}{v \cdot v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44999998807907104:\\ \;\;\;\;\frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{v}}{e^{\frac{1}{v}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + \frac{0.3333333333333333}{v \cdot v}} - \frac{sinTheta_i}{v \cdot v} \cdot \frac{cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)}{2 + \frac{0.3333333333333333}{v \cdot v}}\\ \end{array} \]

Alternative 7: 63.6% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + \frac{0.3333333333333333}{v \cdot v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ cosTheta_i v) (/ cosTheta_O (+ 2.0 (/ 0.3333333333333333 (* v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i / v) * (cosTheta_O / (2.0f + (0.3333333333333333f / (v * 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 = (costheta_i / v) * (costheta_o / (2.0e0 + (0.3333333333333333e0 / (v * v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i / v) * Float32(cosTheta_O / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / v) * (cosTheta_O / (single(2.0) + (single(0.3333333333333333) / (v * v))));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in v around inf 64.5%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 64.5%

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

   2. metadata-eval 64.5%

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

   3. unpow2 64.5%

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

4. Simplified 64.5%

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

5. Taylor expanded in sinTheta_i around 0 64.5%

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]
6. Step-by-step derivation

   1. times-frac 64.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]

   2. unpow2 64.5%

      \[\leadsto \frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + 0.3333333333333333 \cdot \frac{1}{\color{blue}{v \cdot v}}} \]

   3. associate-*r/ 64.5%

      \[\leadsto \frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{v \cdot v}}} \]

   4. metadata-eval 64.5%

      \[\leadsto \frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + \frac{\color{blue}{0.3333333333333333}}{v \cdot v}} \]

7. Simplified 64.5%

   \[\leadsto \color{blue}{\frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

8. Final simplification 64.5%

   \[\leadsto \frac{cosTheta_i}{v} \cdot \frac{cosTheta_O}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

Alternative 8: 58.3% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{1}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ 1.0 (/ v (* cosTheta_i cosTheta_O)))))

C

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

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 = 0.5e0 * (1.0e0 / (v / (costheta_i * costheta_o)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (single(1.0) / (v / (cosTheta_i * cosTheta_O)));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.4%

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

   3. associate-/l* 98.4%

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

   4. associate-/r/ 98.3%

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

   5. *-commutative 98.3%

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

   6. *-commutative 98.3%

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

   7. associate-*r* 98.3%

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

   8. associate-/l/ 98.3%

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

   9. exp-neg 98.3%

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

   10. associate-/l/ 98.3%

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

   11. associate-/r* 98.3%

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

   12. metadata-eval 98.3%

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

   13. associate-*l/ 98.3%

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

   14. *-commutative 98.3%

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

   15. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 58.9%

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

   1. associate-*r/ 58.9%

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

   2. *-commutative 58.9%

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

   3. clear-num 59.4%

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

6. Applied egg-rr 59.4%

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

7. Final simplification 59.4%

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

Alternative 9: 58.3% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}} \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 1.0 (/ (/ v cosTheta_O) cosTheta_i)) 0.5))

C

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

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 = (1.0e0 / ((v / costheta_o) / costheta_i)) * 0.5e0
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(1.0) / ((v / cosTheta_O) / cosTheta_i)) * single(0.5);
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.4%

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

   3. associate-/l* 98.4%

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

   4. associate-/r/ 98.3%

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

   5. *-commutative 98.3%

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

   6. *-commutative 98.3%

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

   7. associate-*r* 98.3%

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

   8. associate-/l/ 98.3%

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

   9. exp-neg 98.3%

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

   10. associate-/l/ 98.3%

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

   11. associate-/r* 98.3%

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

   12. metadata-eval 98.3%

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

   13. associate-*l/ 98.3%

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

   14. *-commutative 98.3%

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

   15. exp-prod 98.3%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 58.9%

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

   1. *-commutative 58.9%

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

   2. associate-/r/ 58.9%

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

   3. clear-num 59.4%

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

6. Applied egg-rr 59.4%

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

7. Final simplification 59.4%

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

Alternative 10: 57.8% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i \cdot cosTheta_O}{v} \cdot 0.5 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ (* cosTheta_i cosTheta_O) v) 0.5))

C

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

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 = ((costheta_i * costheta_o) / v) * 0.5e0
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) * single(0.5);
end

Derivation

1. Initial program 98.5%

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

   1. associate-*l/ 98.4%

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

   2. times-frac 98.5%

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

   3. exp-neg 98.5%

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

   4. associate-*l/ 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate-/l/ 98.5%

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

   7. associate-*l* 98.5%

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

   8. associate-*l* 98.5%

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

   9. *-commutative 98.5%

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

   10. *-commutative 98.5%

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

   11. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 58.9%

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

5. Final simplification 58.9%

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

Alternative 11: 57.9% accurate, 31.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* (* cosTheta_i cosTheta_O) 0.5) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) * 0.5f) / 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 = ((costheta_i * costheta_o) * 0.5e0) / v
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) * single(0.5)) / v;
end

Derivation

1. Initial program 98.5%

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

   1. associate-*l/ 98.4%

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

   2. times-frac 98.5%

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

   3. exp-neg 98.5%

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

   4. associate-*l/ 98.5%

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

   5. *-lft-identity 98.5%

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

   6. associate-/l/ 98.5%

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

   7. associate-*l* 98.5%

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

   8. associate-*l* 98.5%

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

   9. *-commutative 98.5%

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

   10. *-commutative 98.5%

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

   11. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 58.9%

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

   1. associate-*r/ 59.0%

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

6. Applied egg-rr 59.0%

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

7. Final simplification 59.0%

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

Reproduce

herbie shell --seed 2023213 
(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)))