HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 15.6s

Alternatives: 13

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

   1. div-inv 98.6%

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

6. Applied egg-rr 98.6%

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

   1. distribute-frac-neg2 98.6%

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

   2. pow-neg 98.7%

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

8. Applied egg-rr 98.7%

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

   1. div-inv 98.7%

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

10. Applied egg-rr 98.7%

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

11. Taylor expanded in sinTheta_i around inf 98.7%

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

12. Final simplification 98.7%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. div-inv 98.5%

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

   2. *-commutative 98.5%

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 3: 98.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

   1. div-inv 98.6%

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

6. Applied egg-rr 98.6%

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

   1. distribute-frac-neg2 98.6%

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

   2. pow-neg 98.7%

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

8. Applied egg-rr 98.7%

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

   1. div-inv 98.7%

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

10. Applied egg-rr 98.7%

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

11. Taylor expanded in sinTheta_i around 0 98.6%

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

    1. +-commutative 98.6%

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

    2. associate-/l* 98.6%

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

13. Simplified 98.6%

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

14. Final simplification 98.6%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

   1. div-inv 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 98.5%

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

   1. mul-1-neg 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-commutative 98.5%

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

   4. distribute-rgt-neg-in 98.5%

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

   5. distribute-neg-frac2 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 5: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. associate-*r/ 98.4%

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

   2. associate-/l/ 98.4%

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

   3. remove-double-neg 98.4%

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

   4. distribute-rgt-neg-out 98.4%

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

   5. distribute-rgt-neg-out 98.4%

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

   6. distribute-lft-neg-in 98.4%

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

   7. associate-*r/ 98.4%

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

   8. associate-/l/ 98.4%

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

   9. associate-*r/ 98.4%

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

3. Simplified 98.3%

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

   1. associate-*r/ 98.4%

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

   2. clear-num 96.3%

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

   3. *-commutative 96.3%

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

6. Applied egg-rr 96.3%

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

   1. *-commutative 96.3%

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

   2. clear-num 98.4%

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

   3. associate-*r/ 98.3%

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

   4. add-exp-log 66.8%

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

8. Applied egg-rr 66.8%

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

9. Taylor expanded in v around -inf 98.3%

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

    1. mul-1-neg 98.3%

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

    2. distribute-neg-frac2 98.3%

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

    3. +-commutative 98.3%

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

    4. associate-/l* 98.3%

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

    5. neg-mul-1 98.3%

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

    6. distribute-rgt-neg-in 98.3%

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

    7. distribute-lft-out 98.3%

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

11. Simplified 98.3%

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

12. Final simplification 98.3%

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

Alternative 6: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. mul-1-neg 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-commutative 98.5%

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

   4. distribute-rgt-neg-in 98.5%

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

   5. distribute-neg-frac2 98.5%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 7: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

   1. div-inv 98.6%

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

6. Applied egg-rr 98.6%

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

   1. distribute-frac-neg2 98.6%

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

   2. pow-neg 98.7%

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

8. Applied egg-rr 98.7%

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

   1. div-inv 98.7%

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

10. Applied egg-rr 98.7%

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

11. Taylor expanded in sinTheta_i around 0 98.3%

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

12. Final simplification 98.3%

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

Alternative 8: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

   1. div-inv 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 98.3%

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

8. Final simplification 98.3%

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

Alternative 9: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in sinTheta_i around 0 98.1%

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

6. Final simplification 98.1%

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

Alternative 10: 59.1% accurate, 24.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (1.0f / (v / (cosTheta_O * 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 * (1.0e0 / (v / (costheta_o * costheta_i)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 58.8%

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

   1. associate-*l/ 58.8%

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

   2. *-commutative 58.8%

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

7. Simplified 58.8%

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

   1. associate-*r/ 98.4%

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

   2. clear-num 96.3%

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

   3. *-commutative 96.3%

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

9. Applied egg-rr 59.1%

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

Alternative 11: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 58.8%

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

   1. associate-*r/ 58.8%

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

   2. *-commutative 58.8%

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

7. Applied egg-rr 58.8%

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

8. Final simplification 58.8%

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

Alternative 12: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 58.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Add Preprocessing

Alternative 13: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. times-frac 98.3%

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

   2. associate-*l/ 98.4%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

5. Taylor expanded in v around inf 58.8%

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

   1. associate-*l/ 58.8%

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

   2. *-commutative 58.8%

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

7. Simplified 58.8%

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

8. Taylor expanded in cosTheta_i around 0 58.8%

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

   1. associate-/l* 58.8%

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

10. Simplified 58.8%

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

Reproduce

herbie shell --seed 2024136 
(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)))