HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.5%

Time: 14.2s

Alternatives: 10

Speedup: N/A×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (*
   (/ (pow (exp sinTheta_i) (/ sinTheta_O v)) v)
   (/ cosTheta_i (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 ((powf(expf(sinTheta_i), (sinTheta_O / v)) / v) * (cosTheta_i / 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 = (((exp(sintheta_i) ** (sintheta_o / v)) / v) * (costheta_i / 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(Float32((exp(sinTheta_i) ^ Float32(sinTheta_O / v)) / v) * Float32(cosTheta_i / 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 = (((exp(sinTheta_i) ^ (sinTheta_O / v)) / v) * (cosTheta_i / 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. 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. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ cosTheta_O (* v 2.0))
  (/
   (* cosTheta_i (/ (exp (* sinTheta_O (/ sinTheta_i v))) v))
   (sinh (/ 1.0 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 * (expf((sinTheta_O * (sinTheta_i / v))) / v)) / sinhf((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_o / (v * 2.0e0)) * ((costheta_i * (exp((sintheta_o * (sintheta_i / v))) / v)) / sinh((1.0e0 / v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O / (v * single(2.0))) * ((cosTheta_i * (exp((sinTheta_O * (sinTheta_i / v))) / v)) / sinh((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-*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. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

   1. associate-*r/ 98.7%

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

10. Applied egg-rr 98.7%

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

11. Taylor expanded in sinTheta_i around inf 98.5%

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

    1. associate-/l* 98.7%

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

    2. associate-/l* 98.7%

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

13. Simplified 98.7%

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

14. Final simplification 98.7%

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

Alternative 3: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O}{v \cdot 2} \cdot \left(2 \cdot \left(cosTheta\_i \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

   1. associate-*r/ 98.7%

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

10. Applied egg-rr 98.7%

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

11. Taylor expanded in sinTheta_i around 0 98.5%

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

    1. rec-exp 98.6%

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

13. Simplified 98.6%

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

    1. div-inv 98.6%

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

    2. sinh-undef 98.6%

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

15. Applied egg-rr 98.6%

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

    1. associate-*r* 98.6%

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

    2. *-commutative 98.6%

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

    3. *-commutative 98.6%

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

17. Simplified 98.6%

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

18. Final simplification 98.6%

    \[\leadsto \frac{cosTheta\_O}{v \cdot 2} \cdot \left(2 \cdot \left(cosTheta\_i \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]
19. Add Preprocessing

Alternative 4: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O}{v \cdot 2} \cdot \frac{\frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right)} \end{array} \]

FPCore

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O / (v * single(2.0))) * ((cosTheta_i / v) / sinh((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-*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. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

   1. associate-*r/ 98.7%

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

10. Applied egg-rr 98.7%

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

11. Taylor expanded in sinTheta_i around 0 98.5%

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

12. Final simplification 98.5%

    \[\leadsto \frac{cosTheta\_O}{v \cdot 2} \cdot \frac{\frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right)} \]
13. Add Preprocessing

Alternative 5: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {\left(\frac{v}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1} \end{array} \]

FPCore

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

C

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

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((v / (cosTheta_i * cosTheta_O)) ^ single(-1.0));
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. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in v around inf 54.1%

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

    1. *-commutative 54.1%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

11. Simplified 54.1%

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

    1. clear-num 54.8%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]

    2. inv-pow 54.8%

       \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{v}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1}} \]

13. Applied egg-rr 54.8%

    \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{v}{cosTheta\_i \cdot cosTheta\_O}\right)}^{-1}} \]
14. Add Preprocessing

Alternative 6: 58.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O + cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)\right)}{v} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * ((cosTheta_i * cosTheta_O) + (cosTheta_O * (cosTheta_i * (sinTheta_O * (sinTheta_i / 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. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in v around inf 54.1%

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

    1. distribute-lft-out 54.1%

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

    2. *-commutative 54.1%

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

    3. associate-/l* 54.1%

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

    4. associate-/l* 54.1%

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

    5. associate-*r/ 54.1%

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

11. Simplified 54.1%

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

Alternative 7: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in v around inf 54.1%

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

    1. *-commutative 54.1%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

11. Simplified 54.1%

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

    1. associate-*r/ 54.1%

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

13. Applied egg-rr 54.1%

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

Alternative 8: 58.0% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot 2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (/ 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 * (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 * (costheta_o / (v * 2.0e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i * 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 * (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. 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. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in v around inf 54.1%

   \[\leadsto \color{blue}{cosTheta\_i} \cdot \frac{cosTheta\_O}{v \cdot 2} \]
10. Add Preprocessing

Alternative 9: 58.0% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in v around inf 54.1%

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

    1. *-commutative 54.1%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

11. Simplified 54.1%

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

    1. associate-*r/ 54.1%

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

    2. *-commutative 54.1%

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

13. Applied egg-rr 54.1%

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

14. Final simplification 54.1%

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

Alternative 10: 58.0% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.5%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-*r* 98.5%

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

   3. times-frac 98.6%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.8%

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

   2. associate-/r* 98.8%

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

8. Simplified 98.8%

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

   1. associate-*r/ 98.7%

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

10. Applied egg-rr 98.7%

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

11. Taylor expanded in sinTheta_i around inf 98.5%

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

    1. associate-/l* 98.7%

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

    2. associate-/l* 98.7%

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

13. Simplified 98.7%

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

14. Taylor expanded in v around inf 54.1%

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

    1. associate-/l* 54.1%

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

16. Simplified 54.1%

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

Reproduce

herbie shell --seed 2024098 
(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)))