HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 17.0s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \frac{-1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{v \cdot 2}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (+
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ -1.0 v))
    0.6931)
   (log (/ 1.0 (* v 2.0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) + (-1.0f / v)) + 0.6931f) + logf((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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) + ((-1.0e0) / v)) + 0.6931e0) + log((1.0e0 / (v * 2.0e0)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) + Float32(Float32(-1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(v * Float32(2.0))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) + (single(-1.0) / v)) + single(0.6931)) + log((single(1.0) / (v * single(2.0))))));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \frac{-1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{v \cdot 2}\right)} \]
4. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \left(\frac{-1}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+ 0.6931 (log (/ 0.5 v)))
   (- (/ -1.0 v) (/ (* sinTheta_i sinTheta_O) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((0.6931f + logf((0.5f / v))) + ((-1.0f / v) - ((sinTheta_i * sinTheta_O) / v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((0.6931e0 + log((0.5e0 / v))) + (((-1.0e0) / v) - ((sintheta_i * sintheta_o) / v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(0.6931) + log(Float32(Float32(0.5) / v))) + Float32(Float32(Float32(-1.0) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(0.6931) + log((single(0.5) / v))) + ((single(-1.0) / v) - ((sinTheta_i * sinTheta_O) / v))));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0 99.7%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

4. Final simplification 99.7%

   \[\leadsto e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \left(\frac{-1}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(\frac{0.6931 + \frac{-1}{v}}{sinTheta\_i} - \frac{sinTheta\_O}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 v)
  (exp
   (* sinTheta_i (- (/ (+ 0.6931 (/ -1.0 v)) sinTheta_i) (/ sinTheta_O v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((sinTheta_i * (((0.6931f + (-1.0f / v)) / sinTheta_i) - (sinTheta_O / v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((sintheta_i * (((0.6931e0 + ((-1.0e0) / v)) / sintheta_i) - (sintheta_o / v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(sinTheta_i * Float32(Float32(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)) / sinTheta_i) - Float32(sinTheta_O / v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((sinTheta_i * (((single(0.6931) + (single(-1.0) / v)) / sinTheta_i) - (sinTheta_O / v))));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. associate--l- 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + 0.6931} \]

   7. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   8. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + 0.6931} \]

   9. fma-define 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

6. Taylor expanded in sinTheta_i around -inf 99.3%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{-1 \cdot \left(sinTheta\_i \cdot \left(-1 \cdot \frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - -1 \cdot \frac{sinTheta\_O}{v}\right)\right)}} \]
7. Step-by-step derivation

   1. associate-*r* 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(-1 \cdot sinTheta\_i\right) \cdot \left(-1 \cdot \frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - -1 \cdot \frac{sinTheta\_O}{v}\right)}} \]

   2. neg-mul-1 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(-sinTheta\_i\right)} \cdot \left(-1 \cdot \frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - -1 \cdot \frac{sinTheta\_O}{v}\right)} \]

   3. distribute-lft-out-- 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(-sinTheta\_i\right) \cdot \color{blue}{\left(-1 \cdot \left(\frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - \frac{sinTheta\_O}{v}\right)\right)}} \]

8. Simplified 99.3%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(-sinTheta\_i\right) \cdot \left(-1 \cdot \left(\frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - \frac{sinTheta\_O}{v}\right)\right)}} \]

9. Final simplification 99.3%

   \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(\frac{0.6931 + \frac{-1}{v}}{sinTheta\_i} - \frac{sinTheta\_O}{v}\right)} \]
10. Add Preprocessing

Alternative 4: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931 + \left(\frac{-1}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (- (/ -1.0 v) (/ (* sinTheta_i sinTheta_O) v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((0.6931f + ((-1.0f / v) - ((sinTheta_i * sinTheta_O) / v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((0.6931e0 + (((-1.0e0) / v) - ((sintheta_i * sintheta_o) / v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(Float32(-1.0) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) + ((single(-1.0) / v) - ((sinTheta_i * sinTheta_O) / v))));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. associate--l- 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + 0.6931} \]

   7. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   8. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + 0.6931} \]

   9. fma-define 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

6. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 + \left(\frac{-1}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \]
7. Add Preprocessing

Alternative 5: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \frac{0.6931 + \frac{-1}{v}}{sinTheta\_i}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (* sinTheta_i (/ (+ 0.6931 (/ -1.0 v)) sinTheta_i)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((sinTheta_i * ((0.6931f + (-1.0f / v)) / sinTheta_i)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((sintheta_i * ((0.6931e0 + ((-1.0e0) / v)) / sintheta_i)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(sinTheta_i * Float32(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)) / sinTheta_i))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((sinTheta_i * ((single(0.6931) + (single(-1.0) / v)) / sinTheta_i)));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. associate--l- 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + 0.6931} \]

   7. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   8. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + 0.6931} \]

   9. fma-define 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

6. Taylor expanded in sinTheta_i around -inf 99.3%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{-1 \cdot \left(sinTheta\_i \cdot \left(-1 \cdot \frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - -1 \cdot \frac{sinTheta\_O}{v}\right)\right)}} \]
7. Step-by-step derivation

   1. associate-*r* 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(-1 \cdot sinTheta\_i\right) \cdot \left(-1 \cdot \frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - -1 \cdot \frac{sinTheta\_O}{v}\right)}} \]

   2. neg-mul-1 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(-sinTheta\_i\right)} \cdot \left(-1 \cdot \frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - -1 \cdot \frac{sinTheta\_O}{v}\right)} \]

   3. distribute-lft-out-- 99.3%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(-sinTheta\_i\right) \cdot \color{blue}{\left(-1 \cdot \left(\frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - \frac{sinTheta\_O}{v}\right)\right)}} \]

8. Simplified 99.3%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(-sinTheta\_i\right) \cdot \left(-1 \cdot \left(\frac{0.6931 - \frac{1}{v}}{sinTheta\_i} - \frac{sinTheta\_O}{v}\right)\right)}} \]

9. Taylor expanded in sinTheta_O around 0 99.1%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{sinTheta\_i \cdot \left(0.6931 \cdot \frac{1}{sinTheta\_i} - \frac{1}{sinTheta\_i \cdot v}\right)}} \]
10. Step-by-step derivation

    1. metadata-eval 99.1%

       \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(0.6931 \cdot \frac{1}{sinTheta\_i} - \frac{\color{blue}{--1}}{sinTheta\_i \cdot v}\right)} \]

    2. distribute-neg-frac 99.1%

       \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(0.6931 \cdot \frac{1}{sinTheta\_i} - \color{blue}{\left(-\frac{-1}{sinTheta\_i \cdot v}\right)}\right)} \]

    3. associate-/l/ 99.1%

       \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(0.6931 \cdot \frac{1}{sinTheta\_i} - \left(-\color{blue}{\frac{\frac{-1}{v}}{sinTheta\_i}}\right)\right)} \]

    4. distribute-neg-frac 99.1%

       \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(0.6931 \cdot \frac{1}{sinTheta\_i} - \color{blue}{\frac{-\frac{-1}{v}}{sinTheta\_i}}\right)} \]

    5. distribute-neg-frac 99.1%

       \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(0.6931 \cdot \frac{1}{sinTheta\_i} - \frac{\color{blue}{\frac{--1}{v}}}{sinTheta\_i}\right)} \]

    6. metadata-eval 99.1%

       \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(0.6931 \cdot \frac{1}{sinTheta\_i} - \frac{\frac{\color{blue}{1}}{v}}{sinTheta\_i}\right)} \]

    7. associate-*r/ 99.1%

       \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(\color{blue}{\frac{0.6931 \cdot 1}{sinTheta\_i}} - \frac{\frac{1}{v}}{sinTheta\_i}\right)} \]

    8. metadata-eval 99.1%

       \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \left(\frac{\color{blue}{0.6931}}{sinTheta\_i} - \frac{\frac{1}{v}}{sinTheta\_i}\right)} \]

    9. div-sub 99.2%

       \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \color{blue}{\frac{0.6931 - \frac{1}{v}}{sinTheta\_i}}} \]

    10. sub-neg 99.2%

        \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \frac{\color{blue}{0.6931 + \left(-\frac{1}{v}\right)}}{sinTheta\_i}} \]

    11. distribute-neg-frac 99.2%

        \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \frac{0.6931 + \color{blue}{\frac{-1}{v}}}{sinTheta\_i}} \]

    12. metadata-eval 99.2%

        \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \frac{0.6931 + \frac{\color{blue}{-1}}{v}}{sinTheta\_i}} \]

11. Simplified 99.2%

    \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{sinTheta\_i \cdot \frac{0.6931 + \frac{-1}{v}}{sinTheta\_i}}} \]

12. Final simplification 99.2%

    \[\leadsto \frac{0.5}{v} \cdot e^{sinTheta\_i \cdot \frac{0.6931 + \frac{-1}{v}}{sinTheta\_i}} \]
13. Add Preprocessing

Alternative 6: 14.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;{e}^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 3.999999999279835e-23)
   (pow E (* cosTheta_O (/ cosTheta_i v)))
   (exp (/ (* sinTheta_i sinTheta_O) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= 3.999999999279835e-23f) {
		tmp = powf(((float) M_E), (cosTheta_O * (cosTheta_i / v)));
	} else {
		tmp = expf(((sinTheta_i * sinTheta_O) / v));
	}
	return tmp;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_O <= Float32(3.999999999279835e-23))
		tmp = Float32(exp(1)) ^ Float32(cosTheta_O * Float32(cosTheta_i / v));
	else
		tmp = exp(Float32(Float32(sinTheta_i * sinTheta_O) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_O <= single(3.999999999279835e-23))
		tmp = single(2.71828182845904523536) ^ (cosTheta_O * (cosTheta_i / v));
	else
		tmp = exp(((sinTheta_i * sinTheta_O) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 4e-23

   1. Initial program 99.8%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto e^{\color{blue}{1 \cdot \left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

      2. exp-prod 99.6%

         \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

      3. associate-+l+ 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)}} \]

      4. sub-div 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)} \]

      5. sub-div 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)} \]

      6. associate-/r* 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)\right)} \]

      7. metadata-eval 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)\right)} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. exp-1-e 99.6%

         \[\leadsto {\color{blue}{e}}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

      2. +-commutative 99.6%

         \[\leadsto {e}^{\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}\right)}} \]

      3. sub-neg 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\color{blue}{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + \left(-1\right)}}{v}\right)} \]

      4. *-commutative 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(\color{blue}{cosTheta\_O \cdot cosTheta\_i} - sinTheta\_i \cdot sinTheta\_O\right) + \left(-1\right)}{v}\right)} \]

      5. *-commutative 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - \color{blue}{sinTheta\_O \cdot sinTheta\_i}\right) + \left(-1\right)}{v}\right)} \]

      6. metadata-eval 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) + \color{blue}{-1}}{v}\right)} \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{{e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) + -1}{v}\right)}} \]

   7. Taylor expanded in cosTheta_O around inf 11.8%

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

      1. associate-/l* 11.8%

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

   9. Simplified 11.8%

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

   if 4e-23 < sinTheta_O

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in cosTheta_i around 0 99.5%

      \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. Taylor expanded in sinTheta_O around inf 14.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 14.1%

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

      2. associate-*r* 14.1%

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

      3. mul-1-neg 14.1%

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

   6. Simplified 14.1%

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

      1. *-commutative 14.1%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 15.4%

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

      4. sqr-neg 15.4%

         \[\leadsto e^{\frac{sinTheta\_i \cdot \sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

      5. sqrt-unprod 15.4%

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

      6. add-sqr-sqrt 15.4%

         \[\leadsto e^{\frac{sinTheta\_i \cdot \color{blue}{sinTheta\_O}}{v}} \]

      7. pow1 15.4%

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

   8. Applied egg-rr 15.4%

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

      1. unpow1 15.4%

         \[\leadsto e^{\frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}} \]

      2. *-commutative 15.4%

         \[\leadsto e^{\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   10. Simplified 15.4%

       \[\leadsto e^{\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;{e}^{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 14.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;{e}^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 3.999999999279835e-23)
   (pow E (/ (* cosTheta_i cosTheta_O) v))
   (exp (/ (* sinTheta_i sinTheta_O) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= 3.999999999279835e-23f) {
		tmp = powf(((float) M_E), ((cosTheta_i * cosTheta_O) / v));
	} else {
		tmp = expf(((sinTheta_i * sinTheta_O) / v));
	}
	return tmp;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_O <= Float32(3.999999999279835e-23))
		tmp = Float32(exp(1)) ^ Float32(Float32(cosTheta_i * cosTheta_O) / v);
	else
		tmp = exp(Float32(Float32(sinTheta_i * sinTheta_O) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_O <= single(3.999999999279835e-23))
		tmp = single(2.71828182845904523536) ^ ((cosTheta_i * cosTheta_O) / v);
	else
		tmp = exp(((sinTheta_i * sinTheta_O) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 4e-23

   1. Initial program 99.8%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto e^{\color{blue}{1 \cdot \left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

      2. exp-prod 99.6%

         \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

      3. associate-+l+ 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)}} \]

      4. sub-div 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)} \]

      5. sub-div 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)} \]

      6. associate-/r* 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)\right)} \]

      7. metadata-eval 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)\right)} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. exp-1-e 99.6%

         \[\leadsto {\color{blue}{e}}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

      2. +-commutative 99.6%

         \[\leadsto {e}^{\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}\right)}} \]

      3. sub-neg 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\color{blue}{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + \left(-1\right)}}{v}\right)} \]

      4. *-commutative 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(\color{blue}{cosTheta\_O \cdot cosTheta\_i} - sinTheta\_i \cdot sinTheta\_O\right) + \left(-1\right)}{v}\right)} \]

      5. *-commutative 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - \color{blue}{sinTheta\_O \cdot sinTheta\_i}\right) + \left(-1\right)}{v}\right)} \]

      6. metadata-eval 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) + \color{blue}{-1}}{v}\right)} \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{{e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) + -1}{v}\right)}} \]

   7. Taylor expanded in cosTheta_O around inf 11.8%

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

   if 4e-23 < sinTheta_O

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in cosTheta_i around 0 99.5%

      \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. Taylor expanded in sinTheta_O around inf 14.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 14.1%

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

      2. associate-*r* 14.1%

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

      3. mul-1-neg 14.1%

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

   6. Simplified 14.1%

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

      1. *-commutative 14.1%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 15.4%

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

      4. sqr-neg 15.4%

         \[\leadsto e^{\frac{sinTheta\_i \cdot \sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

      5. sqrt-unprod 15.4%

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

      6. add-sqr-sqrt 15.4%

         \[\leadsto e^{\frac{sinTheta\_i \cdot \color{blue}{sinTheta\_O}}{v}} \]

      7. pow1 15.4%

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

   8. Applied egg-rr 15.4%

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

      1. unpow1 15.4%

         \[\leadsto e^{\frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}} \]

      2. *-commutative 15.4%

         \[\leadsto e^{\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   10. Simplified 15.4%

       \[\leadsto e^{\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;{e}^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 14.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_O 3.999999999279835e-23)
   (exp (* cosTheta_O (/ cosTheta_i v)))
   (exp (/ (* sinTheta_i sinTheta_O) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_O <= 3.999999999279835e-23f) {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	} else {
		tmp = expf(((sinTheta_i * sinTheta_O) / 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) :: tmp
    if (sintheta_o <= 3.999999999279835e-23) then
        tmp = exp((costheta_o * (costheta_i / v)))
    else
        tmp = exp(((sintheta_i * sintheta_o) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_O <= Float32(3.999999999279835e-23))
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	else
		tmp = exp(Float32(Float32(sinTheta_i * sinTheta_O) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_O <= single(3.999999999279835e-23))
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	else
		tmp = exp(((sinTheta_i * sinTheta_O) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 4e-23

   1. Initial program 99.8%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto e^{\color{blue}{1 \cdot \left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

      2. exp-prod 99.6%

         \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

      3. associate-+l+ 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)}} \]

      4. sub-div 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)} \]

      5. sub-div 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)} \]

      6. associate-/r* 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)\right)} \]

      7. metadata-eval 99.6%

         \[\leadsto {\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)\right)} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. exp-1-e 99.6%

         \[\leadsto {\color{blue}{e}}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

      2. +-commutative 99.6%

         \[\leadsto {e}^{\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}\right)}} \]

      3. sub-neg 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\color{blue}{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + \left(-1\right)}}{v}\right)} \]

      4. *-commutative 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(\color{blue}{cosTheta\_O \cdot cosTheta\_i} - sinTheta\_i \cdot sinTheta\_O\right) + \left(-1\right)}{v}\right)} \]

      5. *-commutative 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - \color{blue}{sinTheta\_O \cdot sinTheta\_i}\right) + \left(-1\right)}{v}\right)} \]

      6. metadata-eval 99.6%

         \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) + \color{blue}{-1}}{v}\right)} \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{{e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) + -1}{v}\right)}} \]

   7. Taylor expanded in cosTheta_O around inf 11.8%

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

      1. e-exp-1 11.8%

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

      2. pow-exp 11.8%

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

      3. *-un-lft-identity 11.8%

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

      4. associate-/l* 11.8%

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

   9. Applied egg-rr 11.8%

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

   if 4e-23 < sinTheta_O

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in cosTheta_i around 0 99.5%

      \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. Taylor expanded in sinTheta_O around inf 14.1%

      \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 14.1%

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

      2. associate-*r* 14.1%

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

      3. mul-1-neg 14.1%

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

   6. Simplified 14.1%

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

      1. *-commutative 14.1%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 15.4%

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

      4. sqr-neg 15.4%

         \[\leadsto e^{\frac{sinTheta\_i \cdot \sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

      5. sqrt-unprod 15.4%

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

      6. add-sqr-sqrt 15.4%

         \[\leadsto e^{\frac{sinTheta\_i \cdot \color{blue}{sinTheta\_O}}{v}} \]

      7. pow1 15.4%

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

   8. Applied egg-rr 15.4%

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

      1. unpow1 15.4%

         \[\leadsto e^{\frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}} \]

      2. *-commutative 15.4%

         \[\leadsto e^{\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   10. Simplified 15.4%

       \[\leadsto e^{\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot {e}^{\left(0.6931 + \frac{-1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (pow E (+ 0.6931 (/ -1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * powf(((float) M_E), (0.6931f + (-1.0f / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * (Float32(exp(1)) ^ Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (single(2.71828182845904523536) ^ (single(0.6931) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. associate--l- 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + 0.6931} \]

   7. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   8. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + 0.6931} \]

   9. fma-define 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

6. Taylor expanded in sinTheta_O around 0 99.5%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931 - \frac{1}{v}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{1 \cdot \left(0.6931 - \frac{1}{v}\right)}} \]

   2. exp-prod 99.5%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(e^{1}\right)}^{\left(0.6931 - \frac{1}{v}\right)}} \]

   3. e-exp-1 99.5%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{e}}^{\left(0.6931 - \frac{1}{v}\right)} \]

8. Applied egg-rr 99.5%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{e}^{\left(0.6931 - \frac{1}{v}\right)}} \]

9. Final simplification 99.5%

   \[\leadsto \frac{0.5}{v} \cdot {e}^{\left(0.6931 + \frac{-1}{v}\right)} \]
10. Add Preprocessing

Alternative 10: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ -1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((0.6931f + (-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 = (0.5e0 / v) * exp((0.6931e0 + ((-1.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.6%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.6%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. associate--l- 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + 0.6931} \]

   7. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   8. associate-/l* 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + 0.6931} \]

   9. fma-define 99.6%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around 0 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

6. Taylor expanded in sinTheta_O around 0 99.5%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931 - \frac{1}{v}}} \]

7. Final simplification 99.5%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]
8. Add Preprocessing

Alternative 11: 13.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((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 = exp((costheta_o * (costheta_i / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 99.8%

      \[\leadsto e^{\color{blue}{1 \cdot \left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. exp-prod 99.5%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   3. associate-+l+ 99.5%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)}} \]

   4. sub-div 99.5%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)} \]

   5. sub-div 99.5%

      \[\leadsto {\left(e^{1}\right)}^{\left(\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)} \]

   6. associate-/r* 99.5%

      \[\leadsto {\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)\right)} \]

   7. metadata-eval 99.5%

      \[\leadsto {\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)\right)} \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]
5. Step-by-step derivation

   1. exp-1-e 99.5%

      \[\leadsto {\color{blue}{e}}^{\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   2. +-commutative 99.5%

      \[\leadsto {e}^{\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}\right)}} \]

   3. sub-neg 99.5%

      \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\color{blue}{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + \left(-1\right)}}{v}\right)} \]

   4. *-commutative 99.5%

      \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(\color{blue}{cosTheta\_O \cdot cosTheta\_i} - sinTheta\_i \cdot sinTheta\_O\right) + \left(-1\right)}{v}\right)} \]

   5. *-commutative 99.5%

      \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - \color{blue}{sinTheta\_O \cdot sinTheta\_i}\right) + \left(-1\right)}{v}\right)} \]

   6. metadata-eval 99.5%

      \[\leadsto {e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) + \color{blue}{-1}}{v}\right)} \]

6. Simplified 99.5%

   \[\leadsto \color{blue}{{e}^{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) + -1}{v}\right)}} \]

7. Taylor expanded in cosTheta_O around inf 10.2%

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

   1. e-exp-1 10.2%

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

   2. pow-exp 10.2%

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

   3. *-un-lft-identity 10.2%

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

   4. associate-/l* 10.2%

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

9. Applied egg-rr 10.2%

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

10. Final simplification 10.2%

    \[\leadsto e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}} \]
11. Add Preprocessing

Alternative 12: 6.4% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0 99.7%

   \[\leadsto \color{blue}{e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

4. Taylor expanded in sinTheta_O around inf 11.1%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
5. Step-by-step derivation

   1. associate-*r/ 11.1%

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

   2. associate-*r* 11.1%

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

   3. mul-1-neg 11.1%

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

6. Simplified 11.1%

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

7. Taylor expanded in sinTheta_O around 0 6.6%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 6.6%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (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))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))