HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 21.0s

Alternatives: 15

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-/l/ 98.5%

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

   2. div-inv 98.7%

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

   3. div-inv 98.7%

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

   4. clear-num 98.6%

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

5. Applied egg-rr 98.6%

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

   1. associate-*r/ 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-*l/ 98.5%

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

   4. *-rgt-identity 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-*l/ 98.5%

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

   7. *-commutative 98.5%

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

7. Simplified 98.5%

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

   1. div-inv 98.7%

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* (* cosTheta_O (/ cosTheta_i v)) (/ 1.0 v))
  (/ 1.0 (- (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 * (cosTheta_i / v)) * (1.0f / v)) * (1.0f / (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 * (costheta_i / v)) * (1.0e0 / v)) * (1.0e0 / (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 * Float32(cosTheta_i / v)) * Float32(Float32(1.0) / v)) * Float32(Float32(1.0) / 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 * (cosTheta_i / v)) * (single(1.0) / v)) * (single(1.0) / (exp((single(1.0) / v)) - exp((single(-1.0) / v))));
end

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-/l/ 98.5%

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

   2. div-inv 98.7%

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

   3. div-inv 98.7%

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

   4. clear-num 98.6%

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

5. Applied egg-rr 98.6%

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

   1. associate-*r/ 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-*l/ 98.5%

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

   4. *-rgt-identity 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-*l/ 98.5%

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

   7. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in sinTheta_i around 0 98.4%

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

   1. rec-exp 98.4%

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

   2. distribute-neg-frac 98.4%

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

   3. metadata-eval 98.4%

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

10. Simplified 98.4%

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

    1. div-inv 98.7%

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

12. Applied egg-rr 98.6%

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

13. Final simplification 98.6%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.4%

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

   3. associate-/l/ 98.4%

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

   4. associate-*r/ 98.4%

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

   5. *-commutative 98.4%

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

   6. /-rgt-identity 98.4%

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

   7. associate-/r/ 98.4%

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

   8. exp-neg 98.4%

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

   9. remove-double-div 98.4%

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

   10. *-commutative 98.4%

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

   11. associate-*l/ 98.4%

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

   12. exp-prod 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in cosTheta_i around 0 98.4%

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

   1. associate-/r* 98.4%

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

   2. associate-/l* 98.5%

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

   3. unpow2 98.5%

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

   4. associate-/l* 98.5%

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

6. Simplified 98.5%

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

7. Final simplification 98.5%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* cosTheta_O cosTheta_i)
  (* (* v 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 * cosTheta_i) / ((v * 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 * costheta_i) / ((v * 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_O * cosTheta_i) / Float32(Float32(v * 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 * cosTheta_i) / ((v * v) * (exp((single(1.0) / v)) - exp((single(-1.0) / v))));
end

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. *-commutative 98.3%

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

   2. rec-exp 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. metadata-eval 98.3%

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

   5. unpow2 98.3%

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

6. Simplified 98.3%

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

7. Final simplification 98.3%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-/l/ 98.5%

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

   2. div-inv 98.7%

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

   3. div-inv 98.7%

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

   4. clear-num 98.6%

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

5. Applied egg-rr 98.6%

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

   1. associate-*r/ 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-*l/ 98.5%

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

   4. *-rgt-identity 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-*l/ 98.5%

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

   7. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. unpow2 98.3%

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

   2. associate-/r* 98.3%

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

   3. *-commutative 98.3%

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

   4. associate-/l* 98.4%

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

   5. associate-*r/ 98.4%

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

   6. rec-exp 98.4%

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

   7. distribute-neg-frac 98.4%

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

   8. metadata-eval 98.4%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ cosTheta_O (/ (* v v) cosTheta_i))
  (- (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 * v) / cosTheta_i)) / (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 * v) / costheta_i)) / (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_O / Float32(Float32(v * v) / cosTheta_i)) / 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 * v) / cosTheta_i)) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)));
end

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-/l/ 98.5%

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

   2. div-inv 98.7%

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

   3. div-inv 98.7%

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

   4. clear-num 98.6%

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

5. Applied egg-rr 98.6%

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

   1. associate-*r/ 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-*l/ 98.5%

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

   4. *-rgt-identity 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-*l/ 98.5%

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

   7. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. unpow2 98.3%

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

   2. associate-/r* 98.3%

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

   3. associate-/l* 98.4%

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

   4. rec-exp 98.4%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

Alternative 7: 98.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/
   (-
    (* cosTheta_O (/ cosTheta_i v))
    (* (/ cosTheta_O v) (/ (* sinTheta_i (* cosTheta_i sinTheta_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 (((cosTheta_O * (cosTheta_i / v)) - ((cosTheta_O / v) * ((sinTheta_i * (cosTheta_i * sinTheta_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 = (((costheta_o * (costheta_i / v)) - ((costheta_o / v) * ((sintheta_i * (costheta_i * sintheta_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(Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) - Float32(Float32(cosTheta_O / v) * Float32(Float32(sinTheta_i * Float32(cosTheta_i * sinTheta_O)) / v))) / 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 = (((cosTheta_O * (cosTheta_i / v)) - ((cosTheta_O / v) * ((sinTheta_i * (cosTheta_i * sinTheta_O)) / v))) / (sinh((single(1.0) / v)) * single(2.0))) / v;
end

Derivation

1. Initial program 98.4%

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

3. Simplified 98.2%

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

4. Taylor expanded in v around inf 98.1%

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

   1. +-commutative 98.1%

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

   2. mul-1-neg 98.1%

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

   3. unsub-neg 98.1%

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

   4. *-commutative 98.1%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. unpow2 98.2%

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

   8. times-frac 98.2%

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

   9. associate-*r* 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 8: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;\frac{cosTheta_O \cdot \frac{cosTheta_i}{v}}{v} \cdot \frac{1}{e^{\frac{1}{v}} + \left(\frac{1}{v} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.30000001192092896)
   (*
    (/ (* cosTheta_O (/ cosTheta_i v)) v)
    (/ 1.0 (+ (exp (/ 1.0 v)) (+ (/ 1.0 v) -1.0))))
   (/
    (* cosTheta_O cosTheta_i)
    (+
     (* 0.016666666666666666 (/ 1.0 (pow v 3.0)))
     (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.30000001192092896f) {
		tmp = ((cosTheta_O * (cosTheta_i / v)) / v) * (1.0f / (expf((1.0f / v)) + ((1.0f / v) + -1.0f)));
	} else {
		tmp = (cosTheta_O * cosTheta_i) / ((0.016666666666666666f * (1.0f / powf(v, 3.0f))) + (((1.0f / v) * 0.3333333333333333f) + (v * 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.30000001192092896e0) then
        tmp = ((costheta_o * (costheta_i / v)) / v) * (1.0e0 / (exp((1.0e0 / v)) + ((1.0e0 / v) + (-1.0e0))))
    else
        tmp = (costheta_o * costheta_i) / ((0.016666666666666666e0 * (1.0e0 / (v ** 3.0e0))) + (((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.30000001192092896))
		tmp = Float32(Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) / v) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(Float32(Float32(1.0) / v) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(cosTheta_O * cosTheta_i) / Float32(Float32(Float32(0.016666666666666666) * Float32(Float32(1.0) / (v ^ Float32(3.0)))) + Float32(Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.30000001192092896))
		tmp = ((cosTheta_O * (cosTheta_i / v)) / v) * (single(1.0) / (exp((single(1.0) / v)) + ((single(1.0) / v) + single(-1.0))));
	else
		tmp = (cosTheta_O * cosTheta_i) / ((single(0.016666666666666666) * (single(1.0) / (v ^ single(3.0)))) + (((single(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.300000012

   1. Initial program 98.1%

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

      1. times-frac 98.2%

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

      2. *-commutative 98.2%

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

      3. associate-/l* 98.4%

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

      4. associate-/l/ 98.3%

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

      5. distribute-neg-frac 98.3%

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

      6. distribute-rgt-neg-out 98.3%

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

      7. associate-*l/ 98.3%

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

   3. Simplified 98.3%

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

      1. associate-/l/ 98.4%

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

      2. div-inv 98.5%

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

      3. div-inv 98.5%

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

      4. clear-num 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. associate-*r/ 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*l/ 98.2%

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

      4. *-rgt-identity 98.2%

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

      5. *-commutative 98.2%

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

      6. associate-*l/ 98.4%

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

      7. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in sinTheta_i around 0 98.4%

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

      1. rec-exp 98.4%

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

      2. distribute-neg-frac 98.4%

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

      3. metadata-eval 98.4%

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

   10. Simplified 98.4%

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

   11. Taylor expanded in v around inf 70.3%

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

   if 0.300000012 < v

   1. Initial program 98.7%

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

      1. times-frac 98.7%

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

      2. exp-neg 98.7%

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

      3. *-commutative 98.7%

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

      4. exp-neg 98.7%

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

      5. distribute-neg-frac 98.7%

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

      6. *-commutative 98.7%

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

      7. distribute-lft-neg-out 98.7%

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

      8. associate-/l* 98.7%

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

      9. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in sinTheta_i around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. rec-exp 98.4%

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

      3. distribute-neg-frac 98.4%

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

      4. metadata-eval 98.4%

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

      5. unpow2 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in v around inf 73.0%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.30000001192092896:\\ \;\;\;\;\frac{cosTheta_O \cdot \frac{cosTheta_i}{v}}{v} \cdot \frac{1}{e^{\frac{1}{v}} + \left(\frac{1}{v} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)}\\ \end{array} \]

Alternative 9: 67.7% accurate, 1.9× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. *-commutative 98.3%

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

   2. rec-exp 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. metadata-eval 98.3%

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

   5. unpow2 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in v around inf 63.9%

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

8. Final simplification 63.9%

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

Alternative 10: 64.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. *-commutative 98.3%

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

   2. rec-exp 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. metadata-eval 98.3%

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

   5. unpow2 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in v around inf 61.0%

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

   1. *-commutative 61.0%

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

   2. fma-def 61.0%

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

   3. associate-*r/ 61.0%

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

   4. metadata-eval 61.0%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\mathsf{fma}\left(v, 2, \frac{\color{blue}{0.3333333333333333}}{v}\right)} \]

9. Simplified 61.0%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}} \]

10. Final simplification 61.0%

    \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \]

Alternative 11: 64.3% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * cosTheta_i) / (((single(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0)));
end

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. *-commutative 98.3%

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

   2. rec-exp 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. metadata-eval 98.3%

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

   5. unpow2 98.3%

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

6. Simplified 98.3%

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

7. Taylor expanded in v around inf 61.0%

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

8. Final simplification 61.0%

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

Alternative 12: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 54.9%

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

   1. *-commutative 54.9%

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

   2. associate-*l/ 54.9%

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

   3. *-commutative 54.9%

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

6. Simplified 54.9%

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

7. Final simplification 54.9%

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

Alternative 13: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 54.9%

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

   1. *-commutative 54.9%

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

   2. associate-*l/ 54.9%

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

   3. *-commutative 54.9%

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

6. Simplified 54.9%

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

7. Taylor expanded in cosTheta_O around 0 54.9%

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

   1. associate-/l* 54.9%

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

9. Simplified 54.9%

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

    1. associate-/r/ 54.9%

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

11. Applied egg-rr 54.9%

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

12. Final simplification 54.9%

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

Alternative 14: 58.6% accurate, 31.4× speedup

Math

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

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(Float32(cosTheta_O * 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 98.4%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 54.9%

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

5. Final simplification 54.9%

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

Alternative 15: 2.5% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O \cdot cosTheta_i}{0} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_O cosTheta_i) 0.0))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O * cosTheta_i) / 0.0f;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o * costheta_i) / 0.0e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * cosTheta_i) / Float32(0.0))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * cosTheta_i) / single(0.0);
end

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. *-commutative 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-/l* 98.5%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. *-commutative 98.3%

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

   2. rec-exp 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. metadata-eval 98.3%

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

   5. unpow2 98.3%

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

6. Simplified 98.3%

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

   1. expm1-log1p-u 98.2%

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

   2. expm1-udef 98.1%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 2.7%

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

   5. frac-times 10.1%

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

   6. metadata-eval 10.1%

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

   7. metadata-eval 10.1%

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

   8. frac-times 2.7%

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

   9. sqrt-unprod 18.3%

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

   10. add-sqr-sqrt 2.7%

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

8. Applied egg-rr 2.7%

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

   1. expm1-def 2.7%

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

   2. expm1-log1p 2.7%

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

   3. +-inverses 2.7%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{0} \cdot \left(v \cdot v\right)} \]

   4. mul0-lft 2.7%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{0}} \]

10. Simplified 2.7%

    \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{0}} \]

11. Final simplification 2.7%

    \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0} \]

Reproduce

herbie shell --seed 2023278 
(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)))