HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 17.6s

Alternatives: 13

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, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. div-inv 98.9%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (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) / Float32(-v))) * Float32(Float32(cosTheta_i * 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 98.7%

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

3. Final simplification 98.7%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(2.0) * Float32(Float32(cosTheta_i / v) / Float32(exp(Float32(Float32(1.0) / v)) - (Float32(exp(1)) ^ Float32(Float32(-1.0) / v))))) * Float32(Float32(cosTheta_O / v) / Float32(2.0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(2.0) * ((cosTheta_i / v) / (exp((single(1.0) / v)) - (single(2.71828182845904523536) ^ (single(-1.0) / v))))) * ((cosTheta_O / v) / single(2.0));
end

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

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%

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

   2. associate-*r* 98.6%

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

   3. times-frac 98.7%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. associate-/r* 98.6%

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

   2. rec-exp 98.6%

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

   3. distribute-neg-frac 98.6%

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

   4. metadata-eval 98.6%

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

9. Simplified 98.6%

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

    1. *-un-lft-identity 98.6%

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

    2. exp-prod 98.6%

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

11. Applied egg-rr 98.6%

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

    1. exp-1-e 98.6%

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

13. Simplified 98.6%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

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%

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

   2. associate-*r* 98.6%

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

   3. times-frac 98.7%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. associate-/r* 98.6%

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

   2. rec-exp 98.6%

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

   3. distribute-neg-frac 98.6%

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

   4. metadata-eval 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

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%

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

   2. associate-*r* 98.6%

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

   3. times-frac 98.7%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. associate-/r* 98.6%

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

   2. rec-exp 98.6%

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

   3. distribute-neg-frac 98.6%

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

   4. metadata-eval 98.6%

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

9. Simplified 98.6%

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

10. Taylor expanded in cosTheta_i around 0 98.6%

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

11. Final simplification 98.6%

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

Alternative 6: 68.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_O}{v}}{2} \cdot \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} + \left(-1 + \frac{1 - \frac{0.5 + \frac{-0.16666666666666666}{v}}{v}}{v}\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (/ cosTheta_O v) 2.0)
  (*
   2.0
   (/
    (/ cosTheta_i v)
    (+
     (exp (/ 1.0 v))
     (+ -1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.16666666666666666 v)) v)) v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O / v) / 2.0f) * (2.0f * ((cosTheta_i / v) / (expf((1.0f / v)) + (-1.0f + ((1.0f - ((0.5f + (-0.16666666666666666f / v)) / v)) / v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_o / v) / 2.0e0) * (2.0e0 * ((costheta_i / v) / (exp((1.0e0 / v)) + ((-1.0e0) + ((1.0e0 - ((0.5e0 + ((-0.16666666666666666e0) / v)) / v)) / v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_O / v) / Float32(2.0)) * Float32(Float32(2.0) * Float32(Float32(cosTheta_i / v) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(Float32(-1.0) + Float32(Float32(Float32(1.0) - Float32(Float32(Float32(0.5) + Float32(Float32(-0.16666666666666666) / v)) / v)) / v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O / v) / single(2.0)) * (single(2.0) * ((cosTheta_i / v) / (exp((single(1.0) / v)) + (single(-1.0) + ((single(1.0) - ((single(0.5) + (single(-0.16666666666666666) / v)) / v)) / v)))));
end

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

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%

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

   2. associate-*r* 98.6%

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

   3. times-frac 98.7%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. associate-/r* 98.6%

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

   2. rec-exp 98.6%

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

   3. distribute-neg-frac 98.6%

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

   4. metadata-eval 98.6%

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

9. Simplified 98.6%

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

10. Taylor expanded in v around -inf 67.0%

    \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \color{blue}{\left(1 + -1 \cdot \frac{1 + -1 \cdot \frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)}}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]
11. Step-by-step derivation

    1. mul-1-neg 67.0%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \left(1 + \color{blue}{\left(-\frac{1 + -1 \cdot \frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)}\right)}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

    2. unsub-neg 67.0%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \color{blue}{\left(1 - \frac{1 + -1 \cdot \frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}}{v}\right)}}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

    3. mul-1-neg 67.0%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \left(1 - \frac{1 + \color{blue}{\left(-\frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}\right)}}{v}\right)}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

    4. unsub-neg 67.0%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \left(1 - \frac{\color{blue}{1 - \frac{0.5 - 0.16666666666666666 \cdot \frac{1}{v}}{v}}}{v}\right)}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

    5. sub-neg 67.0%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{\color{blue}{0.5 + \left(-0.16666666666666666 \cdot \frac{1}{v}\right)}}{v}}{v}\right)}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

    6. associate-*r/ 67.0%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{0.5 + \left(-\color{blue}{\frac{0.16666666666666666 \cdot 1}{v}}\right)}{v}}{v}\right)}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

    7. metadata-eval 67.0%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{0.5 + \left(-\frac{\color{blue}{0.16666666666666666}}{v}\right)}{v}}{v}\right)}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

    8. distribute-neg-frac 67.0%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{0.5 + \color{blue}{\frac{-0.16666666666666666}{v}}}{v}}{v}\right)}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

    9. metadata-eval 67.0%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \left(1 - \frac{1 - \frac{0.5 + \frac{\color{blue}{-0.16666666666666666}}{v}}{v}}{v}\right)}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

12. Simplified 67.0%

    \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} - \color{blue}{\left(1 - \frac{1 - \frac{0.5 + \frac{-0.16666666666666666}{v}}{v}}{v}\right)}}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]

13. Final simplification 67.0%

    \[\leadsto \frac{\frac{cosTheta\_O}{v}}{2} \cdot \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} + \left(-1 + \frac{1 - \frac{0.5 + \frac{-0.16666666666666666}{v}}{v}}{v}\right)}\right) \]
14. Add Preprocessing

Alternative 7: 70.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{cosTheta\_O}{v}}{2}\\ \mathbf{if}\;v \leq 0.43050000071525574:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} + \left(\frac{1}{v} + -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{\frac{2 + \frac{0.3333333333333333}{{v}^{2}}}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (/ cosTheta_O v) 2.0)))
   (if (<= v 0.43050000071525574)
     (*
      t_0
      (* 2.0 (/ (/ cosTheta_i v) (+ (exp (/ 1.0 v)) (+ (/ 1.0 v) -1.0)))))
     (*
      t_0
      (*
       2.0
       (/
        (/ cosTheta_i v)
        (/ (+ 2.0 (/ 0.3333333333333333 (pow v 2.0))) v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_O / v) / 2.0f;
	float tmp;
	if (v <= 0.43050000071525574f) {
		tmp = t_0 * (2.0f * ((cosTheta_i / v) / (expf((1.0f / v)) + ((1.0f / v) + -1.0f))));
	} else {
		tmp = t_0 * (2.0f * ((cosTheta_i / v) / ((2.0f + (0.3333333333333333f / powf(v, 2.0f))) / v)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (costheta_o / v) / 2.0e0
    if (v <= 0.43050000071525574e0) then
        tmp = t_0 * (2.0e0 * ((costheta_i / v) / (exp((1.0e0 / v)) + ((1.0e0 / v) + (-1.0e0)))))
    else
        tmp = t_0 * (2.0e0 * ((costheta_i / v) / ((2.0e0 + (0.3333333333333333e0 / (v ** 2.0e0))) / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_O / v) / Float32(2.0))
	tmp = Float32(0.0)
	if (v <= Float32(0.43050000071525574))
		tmp = Float32(t_0 * Float32(Float32(2.0) * Float32(Float32(cosTheta_i / v) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(Float32(Float32(1.0) / v) + Float32(-1.0))))));
	else
		tmp = Float32(t_0 * Float32(Float32(2.0) * Float32(Float32(cosTheta_i / v) / Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / (v ^ Float32(2.0)))) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_O / v) / single(2.0);
	tmp = single(0.0);
	if (v <= single(0.43050000071525574))
		tmp = t_0 * (single(2.0) * ((cosTheta_i / v) / (exp((single(1.0) / v)) + ((single(1.0) / v) + single(-1.0)))));
	else
		tmp = t_0 * (single(2.0) * ((cosTheta_i / v) / ((single(2.0) + (single(0.3333333333333333) / (v ^ single(2.0)))) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.430500001

   1. Initial program 98.3%

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

      1. associate-*r/ 98.3%

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

      2. associate-/l/ 98.1%

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

      3. remove-double-neg 98.1%

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

      4. distribute-rgt-neg-out 98.1%

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

      5. distribute-rgt-neg-out 98.1%

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

      6. distribute-lft-neg-in 98.1%

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

      7. associate-*r/ 98.1%

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

      8. associate-/l/ 98.3%

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

      9. associate-*r/ 98.3%

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

   3. Simplified 98.4%

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

      1. associate-*r* 98.4%

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

      2. associate-*r* 98.4%

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

      3. times-frac 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in sinTheta_i around 0 98.3%

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

      1. associate-/r* 98.4%

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

      2. rec-exp 98.4%

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

      3. distribute-neg-frac 98.4%

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

      4. metadata-eval 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in v around inf 68.9%

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

   if 0.430500001 < v

   1. Initial program 99.0%

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

      1. associate-*r/ 99.0%

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

      2. associate-/l/ 99.2%

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

      3. remove-double-neg 99.2%

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

      4. distribute-rgt-neg-out 99.2%

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

      5. distribute-rgt-neg-out 99.2%

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

      6. distribute-lft-neg-in 99.2%

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

      7. associate-*r/ 99.2%

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

      8. associate-/l/ 99.0%

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

      9. associate-*r/ 99.0%

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

   3. Simplified 98.9%

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

      1. associate-*r* 98.9%

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

      2. associate-*r* 98.9%

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

      3. times-frac 99.2%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in sinTheta_i around 0 98.9%

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

      1. associate-/r* 98.8%

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

      2. rec-exp 98.8%

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

      3. distribute-neg-frac 98.8%

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

      4. metadata-eval 98.8%

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

   9. Simplified 98.8%

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

   10. Taylor expanded in v around inf 68.7%

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

       1. associate-*r/ 68.7%

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

       2. metadata-eval 68.7%

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

   12. Simplified 68.7%

       \[\leadsto \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{\color{blue}{\frac{2 + \frac{0.3333333333333333}{{v}^{2}}}{v}}}\right) \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.43050000071525574:\\ \;\;\;\;\frac{\frac{cosTheta\_O}{v}}{2} \cdot \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{e^{\frac{1}{v}} + \left(\frac{1}{v} + -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{cosTheta\_O}{v}}{2} \cdot \left(2 \cdot \frac{\frac{cosTheta\_i}{v}}{\frac{2 + \frac{0.3333333333333333}{{v}^{2}}}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

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%

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

   2. associate-*r* 98.6%

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

   3. times-frac 98.7%

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

6. Applied egg-rr 98.6%

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

7. Taylor expanded in sinTheta_i around 0 98.6%

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

   1. associate-/r* 98.6%

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

   2. rec-exp 98.6%

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

   3. distribute-neg-frac 98.6%

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

   4. metadata-eval 98.6%

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

9. Simplified 98.6%

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

10. Taylor expanded in v around inf 63.2%

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

    1. associate-*r/ 63.2%

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

    2. metadata-eval 63.2%

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

12. Simplified 63.2%

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

13. Final simplification 63.2%

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

Alternative 9: 59.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 57.0%

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

   1. clear-num 57.3%

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

   2. inv-pow 57.3%

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

   3. +-commutative 57.3%

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

   4. *-commutative 57.3%

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

   5. fma-define 57.3%

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

   6. associate-/l* 57.3%

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

7. Applied egg-rr 57.3%

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

   1. unpow-1 57.3%

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

   2. times-frac 57.3%

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

9. Simplified 57.3%

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

Alternative 10: 59.2% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 57.0%

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

6. Taylor expanded in v around inf 57.0%

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

   1. associate-/l* 57.0%

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

8. Simplified 57.0%

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

9. Taylor expanded in cosTheta_O around 0 57.0%

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

    1. *-commutative 57.0%

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

    2. associate-*r/ 57.0%

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

11. Simplified 57.0%

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

    1. associate-*r/ 57.0%

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

    2. clear-num 57.3%

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

13. Applied egg-rr 57.3%

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

Alternative 11: 58.8% accurate, 27.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) * -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_i * costheta_o) * (-0.5e0)) / -v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

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. Taylor expanded in v around -inf 55.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.5 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right) + -0.5 \cdot \frac{-0.5 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right)\right)\right) - -0.16666666666666666 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}{v} + -0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v}} \]

6. Taylor expanded in v around inf 57.0%

   \[\leadsto -1 \cdot \frac{\color{blue}{-0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}}{v} \]

7. Final simplification 57.0%

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

Alternative 12: 58.8% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

   1. times-frac 98.8%

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

   2. *-commutative 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in v around inf 57.0%

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

   1. associate-*r/ 57.0%

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

   2. associate-*r* 57.0%

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

9. Simplified 57.0%

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

10. Final simplification 57.0%

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

Alternative 13: 58.8% 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 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in v around inf 57.0%

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

6. Taylor expanded in v around inf 57.0%

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

   1. associate-/l* 57.0%

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

8. Simplified 57.0%

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

Reproduce

herbie shell --seed 2024103 
(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)))