HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 16.3s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (/ (pow (exp (/ sinTheta_i v)) (- sinTheta_O)) v)
   (* cosTheta_i (/ cosTheta_O 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 ((powf(expf((sinTheta_i / v)), -sinTheta_O) / v) * (cosTheta_i * (cosTheta_O / 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_i / v)) ** -sintheta_o) / v) * (costheta_i * (costheta_o / v))) / (sinh((1.0e0 / v)) * 2.0e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32((exp(Float32(sinTheta_i / v)) ^ Float32(-sinTheta_O)) / v) * Float32(cosTheta_i * Float32(cosTheta_O / 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_i / v)) ^ -sinTheta_O) / v) * (cosTheta_i * (cosTheta_O / 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. *-commutative 98.5%

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

   2. times-frac 98.2%

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

   3. distribute-neg-frac 98.2%

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

   4. distribute-rgt-neg-out 98.2%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. associate-*l/ 98.2%

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

3. Simplified 98.2%

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

   1. associate-*r/ 98.8%

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

   2. exp-prod 98.8%

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

   3. *-commutative 98.8%

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

5. Applied egg-rr 98.8%

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

6. Final simplification 98.8%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i (- sinTheta_O)) v))
   (* (/ 1.0 v) (* cosTheta_i cosTheta_O)))
  (* 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_i * -sinTheta_O) / v)) * ((1.0f / v) * (cosTheta_i * cosTheta_O))) / (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_i * -sintheta_o) / v)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / (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_i * Float32(-sinTheta_O)) / v)) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O))) / 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_i * -sinTheta_O) / v)) * ((single(1.0) / v) * (cosTheta_i * cosTheta_O))) / (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 98.7%

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

   2. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / sinh((single(1.0) / v))) * ((v ^ single(-2.0)) * (cosTheta_O / 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. *-commutative 98.5%

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

   2. times-frac 98.2%

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

   3. distribute-neg-frac 98.2%

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

   4. distribute-rgt-neg-out 98.2%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. associate-*l/ 98.2%

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

3. Simplified 98.2%

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

   1. associate-*r/ 98.8%

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

   2. exp-prod 98.8%

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

   3. *-commutative 98.8%

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

5. Applied egg-rr 98.8%

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

6. Taylor expanded in sinTheta_i around 0 98.4%

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

   1. div-inv 98.4%

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

   2. frac-times 98.3%

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

   3. *-commutative 98.3%

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

   4. *-un-lft-identity 98.3%

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

   5. *-commutative 98.3%

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

   6. associate-*l* 98.3%

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

   7. unpow2 98.3%

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

   8. associate-*r* 98.3%

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

   9. times-frac 98.5%

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

   10. *-commutative 98.5%

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

   11. *-un-lft-identity 98.5%

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

   12. associate-*r* 98.5%

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

   13. unpow2 98.5%

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

   14. times-frac 98.5%

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

   15. pow-flip 98.5%

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

   16. metadata-eval 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 4: 98.5% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* (/ 1.0 v) (* cosTheta_O (/ cosTheta_i 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 ((1.0f / v) * (cosTheta_O * (cosTheta_i / 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 = ((1.0e0 / v) * (costheta_o * (costheta_i / v))) / (sinh((1.0e0 / v)) * 2.0e0)
end function

Julia

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. times-frac 98.2%

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

   3. distribute-neg-frac 98.2%

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

   4. distribute-rgt-neg-out 98.2%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. associate-*l/ 98.2%

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

3. Simplified 98.2%

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

   1. associate-*r/ 98.8%

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

   2. exp-prod 98.8%

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

   3. *-commutative 98.8%

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

5. Applied egg-rr 98.8%

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

6. Taylor expanded in sinTheta_i around 0 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. unpow2 98.4%

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

   3. times-frac 98.5%

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

   4. associate-*l/ 98.5%

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

   5. associate-/r/ 98.5%

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

   6. div-inv 98.5%

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

   7. clear-num 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 5: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O / (v * sinh((single(1.0) / v)))) * (cosTheta_i / (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. *-commutative 98.5%

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

   2. times-frac 98.2%

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

   3. distribute-neg-frac 98.2%

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

   4. distribute-rgt-neg-out 98.2%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. associate-*l/ 98.2%

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

3. Simplified 98.2%

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

   1. associate-*r/ 98.8%

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

   2. exp-prod 98.8%

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

   3. *-commutative 98.8%

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

5. Applied egg-rr 98.8%

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

6. Taylor expanded in sinTheta_i around 0 98.4%

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

   1. div-inv 98.4%

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

   2. frac-times 98.3%

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

   3. *-commutative 98.3%

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

   4. *-un-lft-identity 98.3%

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

   5. *-commutative 98.3%

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

   6. associate-*l* 98.3%

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

   7. unpow2 98.3%

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

   8. associate-*r* 98.3%

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

   9. *-commutative 98.3%

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

   10. associate-*r* 98.3%

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

   11. times-frac 98.3%

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

   12. *-commutative 98.3%

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

8. Applied egg-rr 98.3%

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

9. Final simplification 98.3%

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

Alternative 6: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / (v * sinh((single(1.0) / v)))) * (cosTheta_O / (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. *-commutative 98.5%

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

   2. times-frac 98.2%

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

   3. distribute-neg-frac 98.2%

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

   4. distribute-rgt-neg-out 98.2%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. associate-*l/ 98.2%

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

3. Simplified 98.2%

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

   1. associate-*r/ 98.8%

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

   2. exp-prod 98.8%

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

   3. *-commutative 98.8%

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

5. Applied egg-rr 98.8%

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

6. Taylor expanded in sinTheta_i around 0 98.4%

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

   1. div-inv 98.4%

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

   2. frac-times 98.3%

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

   3. *-commutative 98.3%

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

   4. associate-*l* 98.3%

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

   5. unpow2 98.3%

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

   6. associate-*r* 98.3%

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

   7. *-commutative 98.3%

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

   8. associate-*r/ 98.3%

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

   9. *-commutative 98.3%

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

   10. div-inv 98.3%

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

   11. associate-*r* 98.3%

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

   12. times-frac 98.4%

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

   13. *-commutative 98.4%

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

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

Alternative 7: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O / v) / 2.0f) * ((cosTheta_i / sinhf((1.0f / 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_o / v) / 2.0e0) * ((costheta_i / sinh((1.0e0 / v))) / v)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O / v) / single(2.0)) * ((cosTheta_i / sinh((single(1.0) / 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. Step-by-step derivation

   1. associate-*r/ 98.5%

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

   2. associate-/l/ 98.5%

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

   3. *-commutative 98.5%

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

   4. /-rgt-identity 98.5%

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

   5. associate-/r/ 98.5%

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

   6. exp-neg 98.5%

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

   7. remove-double-div 98.5%

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

   8. *-commutative 98.5%

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

   9. associate-*l/ 98.5%

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

   10. exp-prod 98.5%

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

   11. associate-*l* 98.5%

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

   12. associate-*l* 98.5%

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

   13. *-commutative 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. add-sqr-sqrt 64.2%

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

   2. pow2 64.2%

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

   3. pow-base-1 64.2%

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

   4. add-log-exp 51.2%

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

   5. div-inv 51.2%

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

   6. metadata-eval 51.2%

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

   7. exp-prod 51.2%

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

   8. pow1 51.2%

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

   9. add-log-exp 64.2%

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

   10. *-commutative 64.2%

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

6. Applied egg-rr 64.2%

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

   1. unpow2 64.2%

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

   2. add-sqr-sqrt 98.3%

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

   3. *-commutative 98.3%

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

   4. associate-*r* 98.3%

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

   5. associate-/r* 98.1%

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

   6. *-commutative 98.1%

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

8. Applied egg-rr 98.1%

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

   1. times-frac 98.1%

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

   2. *-commutative 98.1%

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

   3. times-frac 98.4%

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

10. Applied egg-rr 98.4%

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

11. Final simplification 98.4%

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

Alternative 8: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O / v) / v) * ((cosTheta_i / 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 = ((costheta_o / v) / v) * ((costheta_i / sinh((1.0e0 / v))) / 2.0e0)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O / v) / v) * ((cosTheta_i / 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. associate-*r/ 98.5%

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

   2. associate-/l/ 98.5%

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

   3. *-commutative 98.5%

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

   4. /-rgt-identity 98.5%

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

   5. associate-/r/ 98.5%

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

   6. exp-neg 98.5%

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

   7. remove-double-div 98.5%

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

   8. *-commutative 98.5%

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

   9. associate-*l/ 98.5%

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

   10. exp-prod 98.5%

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

   11. associate-*l* 98.5%

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

   12. associate-*l* 98.5%

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

   13. *-commutative 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. add-sqr-sqrt 64.2%

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

   2. pow2 64.2%

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

   3. pow-base-1 64.2%

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

   4. add-log-exp 51.2%

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

   5. div-inv 51.2%

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

   6. metadata-eval 51.2%

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

   7. exp-prod 51.2%

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

   8. pow1 51.2%

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

   9. add-log-exp 64.2%

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

   10. *-commutative 64.2%

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

6. Applied egg-rr 64.2%

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

   1. unpow2 64.2%

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

   2. add-sqr-sqrt 98.3%

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

   3. *-commutative 98.3%

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

   4. associate-*r* 98.3%

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

   5. associate-/r* 98.1%

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

   6. *-commutative 98.1%

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

8. Applied egg-rr 98.1%

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

   1. times-frac 98.1%

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

   2. times-frac 98.5%

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

10. Applied egg-rr 98.5%

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

11. Final simplification 98.5%

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

Alternative 9: 58.7% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0 / ((v / costheta_i) / (costheta_o * 0.5e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.2%

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

   1. associate-/l* 57.2%

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

6. Simplified 57.2%

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

   1. associate-*r/ 57.2%

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

   2. clear-num 57.7%

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

   3. *-commutative 57.7%

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

8. Applied egg-rr 57.7%

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

9. Final simplification 57.7%

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

Alternative 10: 58.1% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o * (costheta_i / v)) * 0.5e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.2%

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

   1. associate-*r/ 57.2%

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

6. Simplified 57.2%

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

7. Final simplification 57.2%

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

Alternative 11: 58.1% accurate, 31.4× speedup

Math

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

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(cosTheta_i * Float32(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%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.2%

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

   1. associate-/l* 57.2%

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

6. Simplified 57.2%

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

   1. associate-/r/ 57.2%

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

8. Applied egg-rr 57.2%

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

9. Final simplification 57.2%

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

Alternative 12: 58.1% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

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

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

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 98.5%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.2%

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

   1. associate-/l* 57.2%

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

6. Simplified 57.2%

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

7. Final simplification 57.2%

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

Alternative 13: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

MATLAB

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 98.5%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 57.2%

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

   1. associate-/l* 57.2%

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

6. Simplified 57.2%

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

   1. clear-num 57.6%

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

   2. un-div-inv 57.6%

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

8. Applied egg-rr 57.6%

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

9. Final simplification 57.6%

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

Reproduce

herbie shell --seed 2023298 
(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)))