HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.4%

Time: 19.6s

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: 98.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* cosTheta_O cosTheta_i)
  (* (pow v 2.0) (- (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) / (powf(v, 2.0f) * (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 ** 2.0e0) * (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(cosTheta_O * cosTheta_i) / Float32((v ^ Float32(2.0)) * 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(2.0)) * (exp((single(1.0) / v)) - exp((single(-1.0) / v))));
end

Derivation

1. Initial program 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 98.7%

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

   1. expm1-log1p-u 98.7%

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

   2. expm1-udef 98.7%

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

   3. rec-exp 98.7%

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

   4. distribute-neg-frac 98.7%

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

   5. metadata-eval 98.7%

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

7. Applied egg-rr 98.7%

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

   1. expm1-def 98.7%

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

   2. expm1-log1p 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

    \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)} \]
11. Add Preprocessing

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 98.6%

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

6. Final simplification 98.6%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l/ 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.24199999868869781:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2} \cdot e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O}{\frac{{v}^{2} \cdot \left(2 \cdot \sinh \mathsf{reciprocal}\left(v\right)\right)}{cosTheta_i}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.24199999868869781)
   (/ (* cosTheta_O cosTheta_i) (* (pow v 2.0) (exp (/ 1.0 v))))
   (/
    cosTheta_O
    (/ (* (pow v 2.0) (* 2.0 (sinh (reciprocal v)))) cosTheta_i))))

Derivation

1. Split input into 2 regimes

2. if v < 0.241999999

   1. Initial program 97.9%

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

      1. associate-*l* 97.9%

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

      2. *-commutative 97.9%

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

      3. times-frac 97.8%

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

      4. *-commutative 97.8%

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

      5. associate-*l/ 97.8%

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

      6. reciprocal-define 79.1%

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

      7. exp-neg 79.1%

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

      8. 1-exp 79.1%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      9. div0 79.1%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      10. exp-diff 79.1%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      11. associate-/l* 79.1%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      12. div-sub 79.1%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      13. neg-sub0 79.1%

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

      14. associate-/r/ 79.1%

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

      15. *-commutative 79.1%

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

   3. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in sinTheta_i around 0 98.1%

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

   6. Taylor expanded in v around inf 81.3%

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

      1. reciprocal-define 81.3%

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

   8. Simplified 81.3%

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

   9. Taylor expanded in v around 0 90.4%

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

   if 0.241999999 < v

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. frac-2neg 98.9%

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

      3. add-sqr-sqrt 57.2%

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

      4. sqrt-unprod 98.9%

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

      5. sqr-neg 98.9%

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

      6. sqrt-unprod 41.7%

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

      7. add-sqr-sqrt 98.9%

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

      8. neg-sub0 98.9%

         \[\leadsto \frac{e^{-\frac{\color{blue}{0 - sinTheta_i}}{-\frac{v}{-sinTheta_O}}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      9. div-sub 98.9%

         \[\leadsto \frac{e^{-\color{blue}{\left(\frac{0}{-\frac{v}{-sinTheta_O}} - \frac{sinTheta_i}{-\frac{v}{-sinTheta_O}}\right)}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      10. add-sqr-sqrt 41.7%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{\color{blue}{\sqrt{-sinTheta_O} \cdot \sqrt{-sinTheta_O}}}} - \frac{sinTheta_i}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      11. sqrt-unprod 98.9%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{\color{blue}{\sqrt{\left(-sinTheta_O\right) \cdot \left(-sinTheta_O\right)}}}} - \frac{sinTheta_i}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      12. sqr-neg 98.9%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{\sqrt{\color{blue}{sinTheta_O \cdot sinTheta_O}}}} - \frac{sinTheta_i}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      13. sqrt-unprod 57.2%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{\color{blue}{\sqrt{sinTheta_O} \cdot \sqrt{sinTheta_O}}}} - \frac{sinTheta_i}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      14. add-sqr-sqrt 98.9%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{\color{blue}{sinTheta_O}}} - \frac{sinTheta_i}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      15. add-sqr-sqrt 48.8%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{sinTheta_O}} - \frac{\color{blue}{\sqrt{sinTheta_i} \cdot \sqrt{sinTheta_i}}}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      16. sqrt-unprod 98.9%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{sinTheta_O}} - \frac{\color{blue}{\sqrt{sinTheta_i \cdot sinTheta_i}}}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      17. sqr-neg 98.9%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{sinTheta_O}} - \frac{\sqrt{\color{blue}{\left(-sinTheta_i\right) \cdot \left(-sinTheta_i\right)}}}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      18. sqrt-unprod 50.1%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{sinTheta_O}} - \frac{\color{blue}{\sqrt{-sinTheta_i} \cdot \sqrt{-sinTheta_i}}}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      19. add-sqr-sqrt 98.9%

          \[\leadsto \frac{e^{-\left(\frac{0}{-\frac{v}{sinTheta_O}} - \frac{\color{blue}{-sinTheta_i}}{-\frac{v}{-sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. Applied egg-rr 98.9%

      \[\leadsto \frac{e^{-\color{blue}{\left(\frac{0}{-\frac{v}{sinTheta_O}} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   5. Step-by-step derivation

      1. div0 98.9%

         \[\leadsto \frac{e^{-\left(\color{blue}{0} - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. neg-sub0 98.9%

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

      3. associate-/r/ 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in sinTheta_i around 0 98.9%

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

   9. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{cosTheta_O}{\frac{\left(2 \cdot \sinh \mathsf{reciprocal}\left(v\right)\right) \cdot {v}^{2}}{cosTheta_i}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.24199999868869781:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2} \cdot e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O}{\frac{{v}^{2} \cdot \left(2 \cdot \sinh \mathsf{reciprocal}\left(v\right)\right)}{cosTheta_i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.45500001311302185:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2} \cdot e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\mathsf{reciprocal}\left(v\right)}^{3} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.45500001311302185)
   (/ (* cosTheta_O cosTheta_i) (* (pow v 2.0) (exp (/ 1.0 v))))
   (/
    (* cosTheta_O cosTheta_i)
    (+
     (* 0.016666666666666666 (pow (reciprocal v) 3.0))
     (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0))))))

Derivation

1. Split input into 2 regimes

2. if v < 0.455000013

   1. Initial program 98.2%

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

      1. associate-*l* 98.2%

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

      2. *-commutative 98.2%

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

      3. times-frac 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*l/ 98.2%

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

      6. reciprocal-define 80.7%

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

      7. exp-neg 80.7%

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

      8. 1-exp 80.7%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      9. div0 80.7%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      10. exp-diff 80.7%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      11. associate-/l* 80.7%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      12. div-sub 80.7%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      13. neg-sub0 80.7%

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

      14. associate-/r/ 80.7%

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

      15. *-commutative 80.7%

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

   3. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in sinTheta_i around 0 98.3%

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

   6. Taylor expanded in v around inf 78.5%

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

      1. reciprocal-define 78.5%

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

   8. Simplified 78.5%

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

   9. Taylor expanded in v around 0 85.4%

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

   if 0.455000013 < v

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. *-commutative 98.9%

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

      3. times-frac 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*l/ 99.0%

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

      6. reciprocal-define 84.5%

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

      7. exp-neg 84.5%

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

      8. 1-exp 84.5%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      9. div0 84.5%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      10. exp-diff 84.5%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      11. associate-/l* 84.5%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      12. div-sub 84.5%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      13. neg-sub0 84.5%

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

      14. associate-/r/ 84.5%

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

      15. *-commutative 84.5%

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

   3. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in sinTheta_i around 0 99.0%

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

   6. Taylor expanded in v around inf 80.5%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{{v}^{3}}} \cdot \sqrt[3]{\frac{1}{{v}^{3}}}\right) \cdot \sqrt[3]{\frac{1}{{v}^{3}}}\right)} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      2. pow3 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{{v}^{3}}}\right)}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      3. cbrt-div 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{{v}^{3}}}\right)}}^{3} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      4. metadata-eval 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\left(\frac{\color{blue}{1}}{\sqrt[3]{{v}^{3}}}\right)}^{3} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      5. rem-cbrt-cube 80.5%

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

      6. reciprocal-define 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\color{blue}{\mathsf{reciprocal}\left(v\right)}}^{3} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   8. Applied egg-rr 80.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.45500001311302185:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2} \cdot e^{\frac{1}{v}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\mathsf{reciprocal}\left(v\right)}^{3} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 0.45500001311302185:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2} \cdot e^{\mathsf{reciprocal}\left(v\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\mathsf{reciprocal}\left(v\right)}^{3} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.45500001311302185)
   (/ (* cosTheta_O cosTheta_i) (* (pow v 2.0) (exp (reciprocal v))))
   (/
    (* cosTheta_O cosTheta_i)
    (+
     (* 0.016666666666666666 (pow (reciprocal v) 3.0))
     (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0))))))

Derivation

1. Split input into 2 regimes

2. if v < 0.455000013

   1. Initial program 98.2%

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

      1. associate-*l* 98.2%

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

      2. *-commutative 98.2%

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

      3. times-frac 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*l/ 98.2%

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

      6. reciprocal-define 80.7%

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

      7. exp-neg 80.7%

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

      8. 1-exp 80.7%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      9. div0 80.7%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      10. exp-diff 80.7%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      11. associate-/l* 80.7%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      12. div-sub 80.7%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      13. neg-sub0 80.7%

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

      14. associate-/r/ 80.7%

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

      15. *-commutative 80.7%

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

   3. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in sinTheta_i around 0 98.3%

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

   6. Taylor expanded in v around inf 78.5%

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

      1. reciprocal-define 78.5%

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

   8. Simplified 78.5%

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

   9. Taylor expanded in v around 0 85.4%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{{v}^{2} \cdot e^{\frac{1}{v}}}} \]
   10. Step-by-step derivation

       1. reciprocal-define 78.9%

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

   11. Simplified 78.9%

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

   if 0.455000013 < v

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. *-commutative 98.9%

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

      3. times-frac 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*l/ 99.0%

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

      6. reciprocal-define 84.5%

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

      7. exp-neg 84.5%

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

      8. 1-exp 84.5%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      9. div0 84.5%

         \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      10. exp-diff 84.5%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

      11. associate-/l* 84.5%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      12. div-sub 84.5%

          \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

      13. neg-sub0 84.5%

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

      14. associate-/r/ 84.5%

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

      15. *-commutative 84.5%

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

   3. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in sinTheta_i around 0 99.0%

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

   6. Taylor expanded in v around inf 80.5%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{{v}^{3}}} \cdot \sqrt[3]{\frac{1}{{v}^{3}}}\right) \cdot \sqrt[3]{\frac{1}{{v}^{3}}}\right)} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      2. pow3 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{{v}^{3}}}\right)}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      3. cbrt-div 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{{v}^{3}}}\right)}}^{3} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      4. metadata-eval 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\left(\frac{\color{blue}{1}}{\sqrt[3]{{v}^{3}}}\right)}^{3} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      5. rem-cbrt-cube 80.5%

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

      6. reciprocal-define 80.5%

         \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\color{blue}{\mathsf{reciprocal}\left(v\right)}}^{3} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   8. Applied egg-rr 80.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.45500001311302185:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{{v}^{2} \cdot e^{\mathsf{reciprocal}\left(v\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\mathsf{reciprocal}\left(v\right)}^{3} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O \cdot cosTheta_i}{\left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right) + 0.016666666666666666 \cdot \mathsf{reciprocal}\left(\left({v}^{3}\right)\right)} \end{array} \]

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))
   (* 0.016666666666666666 (reciprocal (pow v 3.0))))))

Derivation

1. Initial program 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 98.7%

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

6. Taylor expanded in v around inf 71.7%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]
7. Step-by-step derivation

   1. reciprocal-define 71.7%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \color{blue}{\mathsf{reciprocal}\left(\left({v}^{3}\right)\right)} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

8. Applied egg-rr 71.7%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \color{blue}{\mathsf{reciprocal}\left(\left({v}^{3}\right)\right)} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

9. Final simplification 71.7%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right) + 0.016666666666666666 \cdot \mathsf{reciprocal}\left(\left({v}^{3}\right)\right)} \]
10. Add Preprocessing

Alternative 8: 70.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\mathsf{reciprocal}\left(v\right)}^{3} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* cosTheta_O cosTheta_i)
  (+
   (* 0.016666666666666666 (pow (reciprocal v) 3.0))
   (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0)))))

Derivation

1. Initial program 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 98.7%

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

6. Taylor expanded in v around inf 71.7%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 71.7%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{{v}^{3}}} \cdot \sqrt[3]{\frac{1}{{v}^{3}}}\right) \cdot \sqrt[3]{\frac{1}{{v}^{3}}}\right)} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   2. pow3 71.7%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{{v}^{3}}}\right)}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   3. cbrt-div 71.7%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{{v}^{3}}}\right)}}^{3} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   4. metadata-eval 71.7%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\left(\frac{\color{blue}{1}}{\sqrt[3]{{v}^{3}}}\right)}^{3} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   5. rem-cbrt-cube 71.7%

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

   6. reciprocal-define 71.7%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\color{blue}{\mathsf{reciprocal}\left(v\right)}}^{3} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

8. Applied egg-rr 71.7%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \color{blue}{{\mathsf{reciprocal}\left(v\right)}^{3}} + \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

9. Final simplification 71.7%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot {\mathsf{reciprocal}\left(v\right)}^{3} + \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)} \]
10. Add Preprocessing

Alternative 9: 70.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(v \cdot 2 + \frac{0.3333333333333333}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* cosTheta_O cosTheta_i)
  (+
   (* 0.016666666666666666 (/ 1.0 (pow v 3.0)))
   (+ (* 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) / ((0.016666666666666666f * (1.0f / powf(v, 3.0f))) + ((v * 2.0f) + (0.3333333333333333f / 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) / ((0.016666666666666666e0 * (1.0e0 / (v ** 3.0e0))) + ((v * 2.0e0) + (0.3333333333333333e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * cosTheta_i) / Float32(Float32(Float32(0.016666666666666666) * Float32(Float32(1.0) / (v ^ Float32(3.0)))) + Float32(Float32(v * Float32(2.0)) + Float32(Float32(0.3333333333333333) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * cosTheta_i) / ((single(0.016666666666666666) * (single(1.0) / (v ^ single(3.0)))) + ((v * single(2.0)) + (single(0.3333333333333333) / v)));
end

Derivation

1. Initial program 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 98.7%

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

6. Taylor expanded in v around inf 71.7%

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

7. Taylor expanded in v around 0 71.7%

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

8. Final simplification 71.7%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{0.016666666666666666 \cdot \frac{1}{{v}^{3}} + \left(v \cdot 2 + \frac{0.3333333333333333}{v}\right)} \]
9. Add Preprocessing

Alternative 10: 64.1% 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 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 98.7%

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

6. Taylor expanded in v around inf 65.7%

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

   1. *-commutative 65.7%

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

   2. fma-def 65.7%

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

   3. associate-*r/ 65.7%

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

   4. metadata-eval 65.7%

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

8. Simplified 65.7%

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

9. Final simplification 65.7%

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

Alternative 11: 64.1% 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 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 98.7%

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

6. Taylor expanded in v around inf 65.7%

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

7. Final simplification 65.7%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} \]
8. Add Preprocessing

Alternative 12: 58.9% accurate, 25.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 60.3%

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

   1. associate-*l/ 60.3%

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

   2. *-commutative 60.3%

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

7. Simplified 60.3%

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

   1. clear-num 60.3%

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

   2. div-inv 60.3%

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

   3. clear-num 60.3%

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

   4. reciprocal-define 60.3%

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

   5. associate-/l/ 60.3%

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

   6. associate-/r* 60.3%

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

9. Applied egg-rr 60.3%

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

10. Final simplification 60.3%

    \[\leadsto 0.5 \cdot \mathsf{reciprocal}\left(\left(\frac{\frac{v}{cosTheta_i}}{cosTheta_O}\right)\right) \]
11. Add Preprocessing

Alternative 13: 58.3% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 60.3%

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

   1. associate-*l/ 60.3%

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

   2. *-commutative 60.3%

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

7. Simplified 60.3%

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

8. Final simplification 60.3%

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

Alternative 14: 58.3% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 60.3%

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

   1. associate-*l/ 60.3%

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

   2. *-commutative 60.3%

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

7. Simplified 60.3%

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

8. Taylor expanded in cosTheta_i around 0 60.3%

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

   1. associate-/l* 60.3%

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

10. Simplified 60.3%

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

11. Final simplification 60.3%

    \[\leadsto 0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}} \]
12. Add Preprocessing

Alternative 15: 58.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*l* 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. reciprocal-define 82.6%

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

   7. exp-neg 82.6%

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

   8. 1-exp 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{\color{blue}{e^{0}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   9. div0 82.6%

      \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\frac{e^{\color{blue}{\frac{0}{\frac{v}{sinTheta_O}}}}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   10. exp-diff 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{\color{blue}{e^{\frac{0}{\frac{v}{sinTheta_O}} - \frac{sinTheta_i \cdot sinTheta_O}{v}}}}{2 \cdot v} \]

   11. associate-/l* 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{0}{\frac{v}{sinTheta_O}} - \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   12. div-sub 82.6%

       \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\color{blue}{\frac{0 - sinTheta_i}{\frac{v}{sinTheta_O}}}}}{2 \cdot v} \]

   13. neg-sub0 82.6%

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

   14. associate-/r/ 82.6%

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

   15. *-commutative 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_O}{v} \cdot cosTheta_i}{\sinh \mathsf{reciprocal}\left(v\right)} \cdot \frac{e^{\frac{-sinTheta_i}{v} \cdot sinTheta_O}}{v \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf 60.3%

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

   1. associate-*l/ 60.3%

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

   2. *-commutative 60.3%

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

7. Simplified 60.3%

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

   1. clear-num 60.3%

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

   2. div-inv 60.3%

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

   3. clear-num 60.3%

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

   4. un-div-inv 60.3%

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

   5. associate-/l/ 60.3%

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

   6. associate-/r* 60.3%

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

9. Applied egg-rr 60.3%

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

10. Final simplification 60.3%

    \[\leadsto \frac{0.5}{\frac{\frac{v}{cosTheta_i}}{cosTheta_O}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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)))