HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 14.6s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

   1. div-inv 98.7%

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

4. Applied egg-rr 98.7%

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

5. Final simplification 98.7%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

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))
  (exp (* sinTheta_O (/ sinTheta_i (- v))))))

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. times-frac 98.7%

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

4. Applied egg-rr 98.7%

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

   1. associate-/l* 98.7%

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

   2. distribute-neg-frac2 98.7%

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

   3. /-rgt-identity 98.7%

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

   4. *-commutative 98.7%

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

6. Applied egg-rr 98.7%

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

   1. associate-/r* 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-/l* 98.7%

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

   5. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r/ 98.6%

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

   2. associate-/l/ 98.5%

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

   3. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

   1. associate-*r/ 98.6%

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

   2. add-sqr-sqrt -0.0%

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

   3. sqrt-unprod 98.5%

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

   4. sqr-neg 98.5%

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

   5. sqrt-unprod 98.5%

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

   6. add-sqr-sqrt 98.5%

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

6. Applied egg-rr 98.5%

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

Alternative 4: 59.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 58.0%

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

   1. clear-num 58.2%

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

   2. inv-pow 58.2%

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

   3. +-commutative 58.2%

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

   4. fma-define 58.2%

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

   5. associate-/l* 58.2%

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

7. Applied egg-rr 58.2%

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

   1. unpow-1 58.2%

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

   2. associate-/l* 58.5%

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

   3. *-commutative 58.5%

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

9. Simplified 58.5%

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

10. Final simplification 58.5%

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

Alternative 5: 59.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r/ 98.6%

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

   2. associate-/l/ 98.5%

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

   3. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

   1. clear-num 92.5%

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

   2. inv-pow 92.5%

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

6. Applied egg-rr 92.6%

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

   1. unpow-1 92.6%

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

   2. associate-/r* 92.6%

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

   3. exp-prod 92.6%

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

   4. exp-neg 92.5%

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

   5. mul-1-neg 92.5%

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

   6. associate-*r* 92.5%

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

   7. exp-prod 92.5%

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

   8. neg-mul-1 92.5%

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

   9. associate-/l* 92.5%

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

8. Simplified 92.5%

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

   1. *-un-lft-identity 92.5%

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

   2. times-frac 92.4%

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

10. Applied egg-rr 92.4%

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

    1. *-lft-identity 92.4%

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

    2. exp-prod 92.4%

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

    3. *-commutative 92.4%

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

    4. distribute-rgt-neg-out 92.4%

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

    5. associate-*l/ 92.4%

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

    6. associate-*r/ 92.4%

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

    7. distribute-rgt-neg-in 92.4%

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

    8. times-frac 92.5%

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

12. Simplified 92.5%

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

13. Taylor expanded in v around inf 58.2%

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

    1. associate-*r/ 58.2%

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

    2. *-commutative 58.2%

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

    3. *-commutative 58.2%

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

    4. times-frac 58.2%

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

15. Simplified 58.2%

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

16. Final simplification 58.2%

    \[\leadsto \frac{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}}{\frac{v}{cosTheta\_i} \cdot \frac{2}{cosTheta\_O}} \]
17. Add Preprocessing

Alternative 6: 59.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r/ 98.6%

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

   2. associate-/l/ 98.5%

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

   3. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

   1. clear-num 92.5%

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

   2. inv-pow 92.5%

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

6. Applied egg-rr 92.6%

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

   1. unpow-1 92.6%

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

   2. associate-/r* 92.6%

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

   3. exp-prod 92.6%

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

   4. exp-neg 92.5%

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

   5. mul-1-neg 92.5%

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

   6. associate-*r* 92.5%

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

   7. exp-prod 92.5%

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

   8. neg-mul-1 92.5%

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

   9. associate-/l* 92.5%

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

8. Simplified 92.5%

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

   1. *-un-lft-identity 92.5%

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

   2. times-frac 92.4%

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

10. Applied egg-rr 92.4%

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

    1. *-lft-identity 92.4%

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

    2. exp-prod 92.4%

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

    3. *-commutative 92.4%

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

    4. distribute-rgt-neg-out 92.4%

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

    5. associate-*l/ 92.4%

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

    6. associate-*r/ 92.4%

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

    7. distribute-rgt-neg-in 92.4%

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

    8. times-frac 92.5%

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

12. Simplified 92.5%

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

13. Taylor expanded in cosTheta_i around 0 92.4%

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

    1. associate-*r/ 92.5%

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

15. Simplified 92.5%

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

16. Taylor expanded in v around inf 58.2%

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

    1. associate-/r* 58.2%

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

18. Simplified 58.2%

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

19. Final simplification 58.2%

    \[\leadsto \frac{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}}{2 \cdot \frac{\frac{v}{cosTheta\_O}}{cosTheta\_i}} \]
20. Add Preprocessing

Alternative 7: 59.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-*r/ 98.6%

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

   2. associate-/l/ 98.5%

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

   3. remove-double-neg 98.5%

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

   4. distribute-rgt-neg-out 98.5%

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

   5. distribute-rgt-neg-out 98.5%

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

   6. distribute-lft-neg-in 98.5%

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

   7. associate-*r/ 98.5%

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

   8. associate-/l/ 98.6%

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

   9. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

   1. clear-num 92.5%

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

   2. inv-pow 92.5%

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

6. Applied egg-rr 92.6%

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

   1. unpow-1 92.6%

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

   2. associate-/r* 92.6%

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

   3. exp-prod 92.6%

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

   4. exp-neg 92.5%

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

   5. mul-1-neg 92.5%

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

   6. associate-*r* 92.5%

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

   7. exp-prod 92.5%

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

   8. neg-mul-1 92.5%

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

   9. associate-/l* 92.5%

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

8. Simplified 92.5%

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

   1. *-un-lft-identity 92.5%

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

   2. times-frac 92.4%

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

10. Applied egg-rr 92.4%

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

    1. *-lft-identity 92.4%

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

    2. exp-prod 92.4%

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

    3. *-commutative 92.4%

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

    4. distribute-rgt-neg-out 92.4%

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

    5. associate-*l/ 92.4%

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

    6. associate-*r/ 92.4%

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

    7. distribute-rgt-neg-in 92.4%

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

    8. times-frac 92.5%

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

12. Simplified 92.5%

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

13. Taylor expanded in v around inf 58.2%

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

14. Final simplification 58.2%

    \[\leadsto \frac{e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}}{2 \cdot \frac{v}{cosTheta\_i \cdot cosTheta\_O}} \]
15. Add Preprocessing

Alternative 8: 59.1% accurate, 14.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* cosTheta_i cosTheta_O)
  (* v (+ 2.0 (* (/ (* sinTheta_i sinTheta_O) 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 + (((sinTheta_i * sinTheta_O) / 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 + (((sintheta_i * sintheta_o) / v) * 2.0e0)))
end function

Julia

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 58.0%

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

6. Final simplification 58.0%

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

Alternative 9: 59.1% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 58.0%

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

6. Taylor expanded in v around inf 58.0%

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

   1. *-commutative 58.0%

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

   2. associate-*r/ 58.0%

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

8. Simplified 58.0%

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

9. Final simplification 58.0%

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

Alternative 10: 59.1% 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 98.6%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 58.0%

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

6. Taylor expanded in v around inf 58.0%

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

   1. *-commutative 58.0%

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

   2. associate-*r/ 58.0%

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

8. Simplified 58.0%

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

9. Taylor expanded in cosTheta_i around 0 58.0%

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

    1. associate-*r/ 92.5%

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

11. Simplified 58.0%

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

Reproduce

herbie shell --seed 2024106 
(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)))