HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 18.5s

Alternatives: 14

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: 98.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (pow (exp sinTheta_i) (/ sinTheta_O (- v)))
  (/ (* cosTheta_O (* (/ cosTheta_i v) (/ 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 powf(expf(sinTheta_i), (sinTheta_O / -v)) * ((cosTheta_O * ((cosTheta_i / v) * (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 = (exp(sintheta_i) ** (sintheta_o / -v)) * ((costheta_o * ((costheta_i / v) * (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((exp(sinTheta_i) ^ Float32(sinTheta_O / Float32(-v))) * Float32(Float32(cosTheta_O * Float32(Float32(cosTheta_i / v) * 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 = (exp(sinTheta_i) ^ (sinTheta_O / -v)) * ((cosTheta_O * ((cosTheta_i / v) * (single(1.0) / v))) / (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

   4. distribute-frac-neg2 98.5%

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

   5. associate-/l* 98.5%

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

   6. exp-prod 98.5%

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

   7. *-commutative 98.5%

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

   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* 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.7%

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

6. Applied egg-rr 98.7%

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

Alternative 2: 98.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

5. Final simplification 98.6%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i sinTheta_O) (- 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_i * sinTheta_O) / -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_i * sintheta_o) / -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_i * sinTheta_O) / 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_i * sinTheta_O) / -v)) * ((single(1.0) / v) * (cosTheta_O * cosTheta_i))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. div-inv 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

   1. associate-/l* 98.6%

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

   2. associate-*l* 98.6%

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

   3. *-commutative 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around inf 98.6%

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

8. Final simplification 98.6%

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

Alternative 5: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

   1. associate-/l* 98.6%

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

   2. associate-*l* 98.6%

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

   3. *-commutative 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 98.4%

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

8. Final simplification 98.4%

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

Alternative 6: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

   4. distribute-frac-neg2 98.5%

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

   5. associate-/l* 98.5%

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

   6. exp-prod 98.5%

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

   7. *-commutative 98.5%

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

   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* 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in sinTheta_i around 0 98.0%

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

8. Final simplification 98.0%

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

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (/ cosTheta_O (* (sinh (/ 1.0 v)) (* 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 / (sinhf((1.0f / v)) * (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 / (sinh((1.0e0 / v)) * (v * (v * 2.0e0))))
end function

Julia

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

   1. associate-/l* 98.6%

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

   2. associate-*l* 98.6%

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

   3. *-commutative 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 97.9%

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

8. Final simplification 97.9%

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

Alternative 8: 98.4% 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 98.5%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

   4. distribute-frac-neg2 98.5%

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

   5. associate-/l* 98.5%

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

   6. exp-prod 98.5%

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

   7. *-commutative 98.5%

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

   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* 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in sinTheta_i around 0 97.9%

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

6. Final simplification 97.9%

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

Alternative 9: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

   1. associate-/l* 98.6%

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

   2. associate-*l* 98.6%

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

   3. *-commutative 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. *-rgt-identity 97.9%

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

   3. associate-*r* 97.9%

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

   4. times-frac 97.8%

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

9. Applied egg-rr 97.8%

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

    1. associate-*l/ 97.8%

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

    2. *-lft-identity 97.8%

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

    3. *-commutative 97.8%

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

11. Simplified 97.8%

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

Alternative 10: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot \left(2 + \frac{0.3333333333333333}{{v}^{2}}\right)} \end{array} \]

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

   1. associate-/l* 98.6%

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

   2. associate-*l* 98.6%

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

   3. *-commutative 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 97.9%

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

8. Taylor expanded in v around inf 63.3%

   \[\leadsto cosTheta\_i \cdot \frac{cosTheta\_O}{\color{blue}{v \cdot \left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right)}} \]
9. Step-by-step derivation

   1. associate-*r/ 63.3%

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

   2. metadata-eval 63.3%

      \[\leadsto cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot \left(2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}\right)} \]

10. Simplified 63.3%

    \[\leadsto cosTheta\_i \cdot \frac{cosTheta\_O}{\color{blue}{v \cdot \left(2 + \frac{0.3333333333333333}{{v}^{2}}\right)}} \]
11. Add Preprocessing

Alternative 11: 59.2% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{v \cdot \frac{-2}{cosTheta\_O \cdot cosTheta\_i}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return -1.0f / (v * (-2.0f / (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 = (-1.0e0) / (v * ((-2.0e0) / (costheta_o * costheta_i)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

   1. clear-num 93.1%

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

   2. inv-pow 93.1%

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

   3. associate-*l* 93.1%

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

   4. *-commutative 93.1%

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

6. Applied egg-rr 93.1%

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

   1. unpow-1 93.1%

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

   2. associate-/l* 93.1%

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

   3. *-commutative 93.1%

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

   4. times-frac 93.1%

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

   5. associate-*r* 93.1%

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

   6. unpow2 93.1%

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

8. Simplified 93.1%

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

9. Taylor expanded in v around -inf 36.3%

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

    1. associate-*r* 36.3%

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

    2. neg-mul-1 36.3%

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

    3. associate-*r/ 36.3%

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

    4. associate-*r* 36.3%

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

    5. *-commutative 36.3%

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

    6. associate-*r/ 36.3%

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

    7. metadata-eval 36.3%

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

    8. *-commutative 36.3%

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

11. Simplified 36.3%

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

12. Taylor expanded in sinTheta_O around 0 58.1%

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

13. Final simplification 58.1%

    \[\leadsto \frac{-1}{v \cdot \frac{-2}{cosTheta\_O \cdot cosTheta\_i}} \]
14. Add Preprocessing

Alternative 12: 59.0% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v \cdot 2}{cosTheta\_O \cdot cosTheta\_i}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / ((v * 2.0f) / (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 = 1.0e0 / ((v * 2.0e0) / (costheta_o * costheta_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

   1. clear-num 93.1%

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

   2. inv-pow 93.1%

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

   3. associate-*l* 93.1%

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

   4. *-commutative 93.1%

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

6. Applied egg-rr 93.1%

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

   1. unpow-1 93.1%

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

   2. associate-/l* 93.1%

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

   3. *-commutative 93.1%

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

   4. times-frac 93.1%

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

   5. associate-*r* 93.1%

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

   6. unpow2 93.1%

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

8. Simplified 93.1%

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

9. Taylor expanded in v around inf 57.9%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{v}{cosTheta\_O \cdot cosTheta\_i}}} \]
10. Step-by-step derivation

    1. associate-*r/ 57.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot v}{cosTheta\_O \cdot cosTheta\_i}}} \]

    2. *-commutative 57.9%

       \[\leadsto \frac{1}{\frac{\color{blue}{v \cdot 2}}{cosTheta\_O \cdot cosTheta\_i}} \]

    3. *-commutative 57.9%

       \[\leadsto \frac{1}{\frac{v \cdot 2}{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}} \]

11. Simplified 57.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{v \cdot 2}{cosTheta\_i \cdot cosTheta\_O}}} \]

12. Final simplification 57.9%

    \[\leadsto \frac{1}{\frac{v \cdot 2}{cosTheta\_O \cdot cosTheta\_i}} \]
13. Add Preprocessing

Alternative 13: 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 98.5%

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

3. Simplified 98.6%

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

   1. clear-num 93.1%

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

   2. inv-pow 93.1%

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

   3. associate-*l* 93.1%

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

   4. *-commutative 93.1%

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

6. Applied egg-rr 93.1%

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

   1. unpow-1 93.1%

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

   2. associate-/l* 93.1%

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

   3. *-commutative 93.1%

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

   4. times-frac 93.1%

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

   5. associate-*r* 93.1%

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

   6. unpow2 93.1%

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

8. Simplified 93.1%

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

9. Taylor expanded in v around -inf 36.3%

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

    1. associate-*r* 36.3%

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

    2. neg-mul-1 36.3%

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

    3. associate-*r/ 36.3%

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

    4. associate-*r* 36.3%

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

    5. *-commutative 36.3%

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

    6. associate-*r/ 36.3%

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

    7. metadata-eval 36.3%

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

    8. *-commutative 36.3%

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

11. Simplified 36.3%

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

12. Taylor expanded in v around inf 57.5%

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

    1. associate-*r/ 57.5%

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

14. Simplified 57.5%

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

15. Final simplification 57.5%

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

Alternative 14: 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 98.5%

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

3. Simplified 98.6%

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

   1. associate-/l* 98.6%

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

   2. associate-*l* 98.6%

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

   3. *-commutative 98.6%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in v around inf 57.5%

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

   1. associate-*r/ 57.5%

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

   2. *-commutative 57.5%

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

9. Simplified 57.5%

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

10. Taylor expanded in cosTheta_i around 0 57.5%

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

    1. associate-/l* 57.5%

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

12. Applied egg-rr 57.5%

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

Reproduce

herbie shell --seed 2024135 
(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)))