HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 18.6s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(\frac{cosTheta\_i}{0.5 \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)} \cdot \left(cosTheta\_O \cdot \left(\frac{1}{v} \cdot \frac{1}{v \cdot 2}\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (exp (* sinTheta_i (/ sinTheta_O (- v))))
  (*
   (/ cosTheta_i (* 0.5 (- (exp (/ 1.0 v)) (exp (/ -1.0 v)))))
   (* cosTheta_O (* (/ 1.0 v) (/ 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))) * ((cosTheta_i / (0.5f * (expf((1.0f / v)) - expf((-1.0f / v))))) * (cosTheta_O * ((1.0f / v) * (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))) * ((costheta_i / (0.5e0 * (exp((1.0e0 / v)) - exp(((-1.0e0) / v))))) * (costheta_o * ((1.0e0 / v) * (1.0e0 / (v * 2.0e0)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(sinTheta_i * Float32(sinTheta_O / Float32(-v)))) * Float32(Float32(cosTheta_i / Float32(Float32(0.5) * Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v))))) * Float32(cosTheta_O * Float32(Float32(Float32(1.0) / v) * Float32(Float32(1.0) / 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))) * ((cosTheta_i / (single(0.5) * (exp((single(1.0) / v)) - exp((single(-1.0) / v))))) * (cosTheta_O * ((single(1.0) / v) * (single(1.0) / (v * single(2.0))))));
end

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac2 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-/l* 98.5%

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

   8. distribute-frac-neg2 98.5%

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

   9. distribute-frac-neg 98.5%

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

   10. associate-*l* 98.5%

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

   11. associate-/l* 98.5%

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

   12. times-frac 98.5%

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

   13. *-commutative 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. times-frac 98.7%

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

6. Applied egg-rr 98.7%

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

   1. div-inv 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in v around 0 98.8%

   \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{\color{blue}{0.5 \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \cdot \left(\frac{cosTheta\_O}{1} \cdot \left(\frac{1}{v} \cdot \frac{1}{v \cdot 2}\right)\right)\right) \]
10. Step-by-step derivation

    1. rec-exp 98.8%

       \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{0.5 \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \cdot \left(\frac{cosTheta\_O}{1} \cdot \left(\frac{1}{v} \cdot \frac{1}{v \cdot 2}\right)\right)\right) \]

    2. distribute-neg-frac 98.8%

       \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{0.5 \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \cdot \left(\frac{cosTheta\_O}{1} \cdot \left(\frac{1}{v} \cdot \frac{1}{v \cdot 2}\right)\right)\right) \]

    3. metadata-eval 98.8%

       \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{0.5 \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \cdot \left(\frac{cosTheta\_O}{1} \cdot \left(\frac{1}{v} \cdot \frac{1}{v \cdot 2}\right)\right)\right) \]

11. Simplified 98.8%

    \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{\color{blue}{0.5 \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \cdot \left(\frac{cosTheta\_O}{1} \cdot \left(\frac{1}{v} \cdot \frac{1}{v \cdot 2}\right)\right)\right) \]

12. Final simplification 98.8%

    \[\leadsto e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \cdot \left(\frac{cosTheta\_i}{0.5 \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)} \cdot \left(cosTheta\_O \cdot \left(\frac{1}{v} \cdot \frac{1}{v \cdot 2}\right)\right)\right) \]
13. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (exp (* sinTheta_i (/ sinTheta_O (- v))))
  (* (/ cosTheta_i (sinh (/ 1.0 v))) (* (/ 1.0 v) (* cosTheta_O (/ 0.5 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 / sinhf((1.0f / v))) * ((1.0f / v) * (cosTheta_O * (0.5f / v))));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac2 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-/l* 98.5%

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

   8. distribute-frac-neg2 98.5%

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

   9. distribute-frac-neg 98.5%

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

   10. associate-*l* 98.5%

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

   11. associate-/l* 98.5%

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

   12. times-frac 98.5%

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

   13. *-commutative 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. times-frac 98.7%

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

6. Applied egg-rr 98.7%

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

   1. div-inv 98.8%

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

8. Applied egg-rr 98.8%

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

   1. /-rgt-identity 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-*l/ 98.7%

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

   4. *-un-lft-identity 98.7%

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

   5. associate-*l/ 98.7%

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

   6. *-commutative 98.7%

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

   7. associate-/r* 98.7%

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

   8. metadata-eval 98.7%

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

10. Applied egg-rr 98.7%

    \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{\frac{0.5}{v} \cdot cosTheta\_O}{v}}\right) \]
11. Step-by-step derivation

    1. div-inv 98.8%

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

12. Applied egg-rr 98.8%

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

13. Final simplification 98.8%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (exp (* sinTheta_i (/ sinTheta_O (- v))))
  (* (/ cosTheta_i (sinh (/ 1.0 v))) (* (/ 0.5 v) (/ cosTheta_O 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 / sinhf((1.0f / v))) * ((0.5f / v) * (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 = exp((sintheta_i * (sintheta_o / -v))) * ((costheta_i / sinh((1.0e0 / v))) * ((0.5e0 / v) * (costheta_o / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac2 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-/l* 98.5%

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

   8. distribute-frac-neg2 98.5%

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

   9. distribute-frac-neg 98.5%

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

   10. associate-*l* 98.5%

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

   11. associate-/l* 98.5%

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

   12. times-frac 98.5%

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

   13. *-commutative 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. times-frac 98.7%

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

6. Applied egg-rr 98.7%

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

   1. div-inv 98.8%

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

8. Applied egg-rr 98.8%

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

   1. /-rgt-identity 98.8%

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

   2. *-commutative 98.8%

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

   3. associate-*l/ 98.7%

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

   4. *-un-lft-identity 98.7%

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

   5. associate-*l/ 98.7%

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

   6. *-commutative 98.7%

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

   7. associate-/r* 98.7%

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

   8. metadata-eval 98.7%

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

10. Applied egg-rr 98.7%

    \[\leadsto e^{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \cdot \left(\frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{\frac{0.5}{v} \cdot cosTheta\_O}{v}}\right) \]
11. Step-by-step derivation

    1. *-commutative 98.7%

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

    2. *-un-lft-identity 98.7%

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

    3. frac-times 98.7%

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

    4. /-rgt-identity 98.7%

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

    5. div-inv 98.7%

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

    6. metadata-eval 98.7%

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

    7. inv-pow 98.7%

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

    8. unpow-prod-down 98.7%

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

    9. *-commutative 98.7%

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

    10. inv-pow 98.7%

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

    11. *-un-lft-identity 98.7%

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

    12. associate-*l/ 98.8%

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

    13. associate-*r* 98.8%

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

    14. div-inv 98.7%

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

    15. inv-pow 98.7%

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

    16. *-commutative 98.7%

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

    17. unpow-prod-down 98.7%

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

    18. metadata-eval 98.7%

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

    19. inv-pow 98.7%

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

    20. div-inv 98.7%

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

12. Applied egg-rr 98.7%

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

13. Final simplification 98.7%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. 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/ 98.6%

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

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. *-commutative 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-*l* 98.5%

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

   4. *-commutative 98.5%

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

   5. times-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r/ 98.5%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 98.4%

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

   10. sqr-neg 98.4%

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

   11. sqrt-unprod 98.4%

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

   12. add-sqr-sqrt 98.4%

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

   13. add-sqr-sqrt 66.5%

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

   14. add-sqr-sqrt 98.4%

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

   15. associate-/l* 98.4%

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

   16. exp-prod 98.4%

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

6. Applied egg-rr 98.4%

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

   1. associate-*l/ 98.7%

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

   2. *-commutative 98.7%

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

8. Simplified 98.7%

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

   1. associate-/l* 98.5%

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

10. Applied egg-rr 98.5%

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

11. Taylor expanded in sinTheta_i around 0 98.3%

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

    1. associate-/r* 98.5%

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

    2. rec-exp 98.5%

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

    3. distribute-neg-frac 98.5%

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

    4. metadata-eval 98.5%

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

13. Simplified 98.5%

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

14. Final simplification 98.5%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{{v}^{2}}}{\sinh \left(\frac{1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (/ (* cosTheta_i cosTheta_O) (pow v 2.0))) (sinh (/ 1.0 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) / powf(v, 2.0f))) / 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 = (0.5e0 * ((costheta_i * costheta_o) / (v ** 2.0e0))) / sinh((1.0e0 / 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) / (v ^ Float32(2.0)))) / sinh(Float32(Float32(1.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * ((cosTheta_i * cosTheta_O) / (v ^ single(2.0)))) / sinh((single(1.0) / v));
end

Derivation

1. Initial program 98.6%

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

   1. associate-*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/ 98.6%

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

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. *-commutative 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-*l* 98.5%

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

   4. *-commutative 98.5%

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

   5. times-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r/ 98.5%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 98.4%

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

   10. sqr-neg 98.4%

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

   11. sqrt-unprod 98.4%

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

   12. add-sqr-sqrt 98.4%

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

   13. add-sqr-sqrt 66.5%

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

   14. add-sqr-sqrt 98.4%

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

   15. associate-/l* 98.4%

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

   16. exp-prod 98.4%

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

6. Applied egg-rr 98.4%

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

   1. associate-*l/ 98.7%

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

   2. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in v around inf 98.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2}}}}{\sinh \left(\frac{1}{v}\right)} \]

10. Final simplification 98.5%

    \[\leadsto \frac{0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{{v}^{2}}}{\sinh \left(\frac{1}{v}\right)} \]
11. Add Preprocessing

Alternative 6: 58.7% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*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/ 98.6%

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

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. *-commutative 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-*l* 98.5%

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

   4. *-commutative 98.5%

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

   5. times-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r/ 98.5%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 98.4%

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

   10. sqr-neg 98.4%

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

   11. sqrt-unprod 98.4%

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

   12. add-sqr-sqrt 98.4%

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

   13. add-sqr-sqrt 66.5%

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

   14. add-sqr-sqrt 98.4%

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

   15. associate-/l* 98.4%

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

   16. exp-prod 98.4%

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

6. Applied egg-rr 98.4%

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

   1. associate-*l/ 98.7%

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

   2. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in v around inf 57.3%

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

    1. associate-/l* 57.3%

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

11. Simplified 57.3%

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

    1. associate-*r/ 57.3%

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

    2. clear-num 57.7%

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

13. Applied egg-rr 57.7%

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

14. Final simplification 57.7%

    \[\leadsto 0.5 \cdot \frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}} \]
15. Add Preprocessing

Alternative 7: 58.2% 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 98.6%

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

   1. associate-*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/ 98.6%

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

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. *-commutative 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-*l* 98.5%

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

   4. *-commutative 98.5%

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

   5. times-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r/ 98.5%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 98.4%

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

   10. sqr-neg 98.4%

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

   11. sqrt-unprod 98.4%

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

   12. add-sqr-sqrt 98.4%

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

   13. add-sqr-sqrt 66.5%

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

   14. add-sqr-sqrt 98.4%

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

   15. associate-/l* 98.4%

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

   16. exp-prod 98.4%

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

6. Applied egg-rr 98.4%

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

   1. associate-*l/ 98.7%

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

   2. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in v around inf 57.3%

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

    1. associate-/l* 57.3%

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

11. Simplified 57.3%

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

12. Final simplification 57.3%

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

Alternative 8: 58.2% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. 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/ 98.6%

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

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. *-commutative 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-*l* 98.5%

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

   4. *-commutative 98.5%

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

   5. times-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r/ 98.5%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 98.4%

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

   10. sqr-neg 98.4%

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

   11. sqrt-unprod 98.4%

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

   12. add-sqr-sqrt 98.4%

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

   13. add-sqr-sqrt 66.5%

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

   14. add-sqr-sqrt 98.4%

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

   15. associate-/l* 98.4%

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

   16. exp-prod 98.4%

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

6. Applied egg-rr 98.4%

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

   1. associate-*l/ 98.7%

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

   2. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in v around inf 57.3%

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

    1. associate-/l* 57.3%

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

11. Simplified 57.3%

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

    1. associate-*r* 57.3%

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

    2. associate-*r/ 57.3%

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

    3. *-commutative 57.3%

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

13. Applied egg-rr 57.3%

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

14. Final simplification 57.3%

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

Reproduce

herbie shell --seed 2024046 
(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)))