HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 15.6s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((sinTheta_O / v) * -sinTheta_i)) * ((cosTheta_i * (cosTheta_O / v)) * (((single(2.0) * sinh((single(1.0) / v))) / (single(1.0) / v)) ^ single(-1.0)));
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. distribute-neg-frac 98.6%

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

   2. *-commutative 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. div-inv 98.6%

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

   2. associate-/l* 98.6%

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

   3. associate-*l/ 98.6%

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

   4. associate-*r/ 98.6%

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

5. Applied egg-rr 98.6%

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

   1. associate-*l* 98.6%

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

   2. associate-/r/ 98.6%

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

   3. associate-/r* 98.8%

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

   4. *-commutative 98.8%

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

7. Simplified 98.8%

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

   1. clear-num 98.8%

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

   2. inv-pow 98.8%

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((sinTheta_O / v) * -sinTheta_i)) * ((cosTheta_i * (cosTheta_O / v)) * ((single(1.0) / 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. distribute-neg-frac 98.6%

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

   2. *-commutative 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. div-inv 98.6%

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

   2. associate-/l* 98.6%

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

   3. associate-*l/ 98.6%

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

   4. associate-*r/ 98.6%

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

5. Applied egg-rr 98.6%

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

   1. associate-*l* 98.6%

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

   2. associate-/r/ 98.6%

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

   3. associate-/r* 98.8%

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

   4. *-commutative 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / (v * (sinh((single(1.0) / v)) * (v * single(2.0))))) * (cosTheta_O / exp((sinTheta_i / (v / sinTheta_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-*l/ 98.6%

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

   2. times-frac 98.6%

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

   3. exp-neg 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-lft-identity 98.6%

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

   6. associate-/l/ 98.6%

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

   7. associate-*l* 98.6%

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

   8. associate-*l* 98.6%

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

   9. *-commutative 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

3. Simplified 98.6%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)\right)} \]

   2. expm1-udef 53.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)} - 1} \]

   3. associate-*l* 53.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}\right)}}\right)} - 1 \]

   4. *-commutative 53.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}\right)}\right)} - 1 \]

   5. exp-prod 53.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot \left(v \cdot 2\right)\right) \cdot \color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}\right)}\right)} - 1 \]

5. Applied egg-rr 53.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}\right)}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 98.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot \left(v \cdot 2\right)\right) \cdot {\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}\right)}\right)\right)} \]

   2. expm1-log1p 98.6%

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

   3. associate-*r* 98.6%

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

   4. exp-prod 98.6%

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

   5. *-commutative 98.6%

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

   6. associate-*r/ 98.6%

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

   7. times-frac 98.7%

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

   8. associate-*r/ 98.7%

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

   9. *-commutative 98.7%

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

   10. exp-prod 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*l/ 98.6%

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

   2. times-frac 98.6%

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

   3. exp-neg 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-lft-identity 98.6%

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

   6. associate-/l/ 98.6%

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

   7. associate-*l* 98.6%

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

   8. associate-*l* 98.6%

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

   9. *-commutative 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

3. Simplified 98.6%

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

   1. expm1-log1p-u 98.3%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

   2. *-commutative 98.3%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

5. Applied egg-rr 98.3%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
6. Step-by-step derivation

   1. times-frac 98.4%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

   2. expm1-log1p-u 98.7%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * ((cosTheta_O / v) * (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. distribute-neg-frac 98.6%

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

   2. *-commutative 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. times-frac 98.5%

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

   3. rec-exp 98.5%

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

   4. distribute-neg-frac 98.5%

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

   5. metadata-eval 98.5%

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

   6. unpow2 98.5%

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

6. Simplified 98.5%

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

   1. associate-/r* 98.3%

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

   2. div-inv 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ cosTheta_O v) (/ cosTheta_i 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_O / v) * (cosTheta_i / 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_o / v) * (costheta_i / 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(Float32(cosTheta_O / v) * Float32(cosTheta_i / 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_O / v) * (cosTheta_i / 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. distribute-neg-frac 98.6%

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

   2. *-commutative 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.4%

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

   1. associate-/r* 98.3%

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

   2. unpow2 98.3%

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

   3. rec-exp 98.2%

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

   4. distribute-neg-frac 98.2%

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

   5. metadata-eval 98.2%

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

6. Simplified 98.2%

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

   1. times-frac 98.4%

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

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

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i / single(2.0)) / sinh((single(1.0) / v))) * (cosTheta_O / (v * 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. distribute-neg-frac 98.6%

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

   2. *-commutative 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. times-frac 98.5%

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

   3. rec-exp 98.5%

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

   4. distribute-neg-frac 98.5%

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

   5. metadata-eval 98.5%

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

   6. unpow2 98.5%

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

6. Simplified 98.5%

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

   1. expm1-log1p-u 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}\right)\right)} \cdot \frac{cosTheta_O}{v \cdot v} \]

   2. expm1-udef 61.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}\right)} - 1\right)} \cdot \frac{cosTheta_O}{v \cdot v} \]

   3. div-inv 61.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\color{blue}{-1 \cdot \frac{1}{v}}}}\right)} - 1\right) \cdot \frac{cosTheta_O}{v \cdot v} \]

   4. neg-mul-1 61.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{cosTheta_i}{e^{\frac{1}{v}} - e^{\color{blue}{-\frac{1}{v}}}}\right)} - 1\right) \cdot \frac{cosTheta_O}{v \cdot v} \]

   5. sinh-undef 61.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{cosTheta_i}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}}\right)} - 1\right) \cdot \frac{cosTheta_O}{v \cdot v} \]

8. Applied egg-rr 61.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta_i}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot \frac{cosTheta_O}{v \cdot v} \]
9. Step-by-step derivation

   1. expm1-def 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_i}{2 \cdot \sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot \frac{cosTheta_O}{v \cdot v} \]

   2. expm1-log1p 98.5%

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

   3. associate-/r* 98.5%

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

10. Simplified 98.5%

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

11. Final simplification 98.5%

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

Alternative 8: 64.6% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{2 + \frac{0.3333333333333333}{v \cdot v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i (/ cosTheta_O v))
  (/ 1.0 (+ 2.0 (/ 0.3333333333333333 (* v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * (cosTheta_O / v)) * (1.0f / (2.0f + (0.3333333333333333f / (v * v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * (costheta_o / v)) * (1.0e0 / (2.0e0 + (0.3333333333333333e0 / (v * v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) * Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * (cosTheta_O / v)) * (single(1.0) / (single(2.0) + (single(0.3333333333333333) / (v * 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. *-commutative 98.6%

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

   2. associate-*r/ 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-commutative 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-*r* 98.6%

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

   8. associate-/l/ 98.6%

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

   9. exp-neg 98.6%

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

   10. associate-/l/ 98.6%

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

   11. associate-/r* 98.6%

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

   12. metadata-eval 98.6%

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

   13. associate-*l/ 98.6%

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

   14. *-commutative 98.6%

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

   15. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in sinTheta_O around 0 98.4%

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

   1. rec-exp 98.4%

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

   2. distribute-neg-frac 98.4%

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

   3. metadata-eval 98.4%

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

6. Simplified 98.4%

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

7. Taylor expanded in v around inf 65.0%

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

   1. associate-*r/ 65.0%

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

   2. metadata-eval 65.0%

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

   3. unpow2 65.0%

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

9. Simplified 65.0%

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

10. Final simplification 65.0%

    \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{1}{2 + \frac{0.3333333333333333}{v \cdot v}} \]

Alternative 9: 59.4% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. distribute-neg-frac 98.6%

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

   2. *-commutative 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in v around inf 59.5%

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

   1. associate-*r/ 59.6%

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

   2. *-commutative 59.6%

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

   3. associate-*r/ 59.5%

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

   4. associate-*l/ 59.6%

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

   5. associate-*l* 59.6%

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

6. Simplified 59.6%

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

   1. associate-*l/ 59.6%

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

   2. clear-num 60.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{v}{cosTheta_i \cdot \left(cosTheta_O \cdot 0.5\right)}}} \]

8. Applied egg-rr 60.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{v}{cosTheta_i \cdot \left(cosTheta_O \cdot 0.5\right)}}} \]

9. Final simplification 60.2%

   \[\leadsto \frac{1}{\frac{v}{cosTheta_i \cdot \left(cosTheta_O \cdot 0.5\right)}} \]

Alternative 10: 59.0% accurate, 31.4× speedup

Math

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

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(0.5) * Float32(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 98.6%

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

   1. distribute-neg-frac 98.6%

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

   2. *-commutative 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in v around inf 59.5%

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

5. Final simplification 59.5%

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

Alternative 11: 59.0% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*r/ 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-commutative 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-*r* 98.6%

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

   8. associate-/l/ 98.6%

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

   9. exp-neg 98.6%

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

   10. associate-/l/ 98.6%

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

   11. associate-/r* 98.6%

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

   12. metadata-eval 98.6%

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

   13. associate-*l/ 98.6%

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

   14. *-commutative 98.6%

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

   15. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in v around inf 59.6%

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

5. Final simplification 59.6%

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

Alternative 12: 59.0% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / v) * (cosTheta_O * single(0.5));
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. distribute-neg-frac 98.6%

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

   2. *-commutative 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in v around inf 59.5%

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

   1. associate-*r/ 59.6%

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

   2. *-commutative 59.6%

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

   3. associate-*r/ 59.5%

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

   4. associate-*l/ 59.6%

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

   5. associate-*l* 59.6%

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

6. Simplified 59.6%

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

7. Final simplification 59.6%

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

Alternative 13: 59.0% 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 98.6%

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

   1. distribute-neg-frac 98.6%

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

   2. *-commutative 98.6%

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

   3. distribute-rgt-neg-in 98.6%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in v around inf 59.5%

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

   1. associate-*r/ 59.6%

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

6. Applied egg-rr 59.6%

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

7. Final simplification 59.6%

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

Reproduce

herbie shell --seed 2023258 
(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)))