HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 12.9s

Alternatives: 12

Speedup: N/A×

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: 99.6% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. exp-sum 99.8%

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

   5. add-exp-log 99.8%

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

   6. associate--r+ 99.8%

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

   7. associate-*r/ 99.8%

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

   8. associate-*r/ 99.8%

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

   9. sub-div 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. add-exp-log 99.8%

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

   3. prod-exp 99.8%

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

8. Applied egg-rr 99.8%

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

   1. exp-sum 99.8%

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

   2. add-exp-log 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*l/ 99.8%

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

   5. clear-num 99.8%

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

10. Applied egg-rr 99.8%

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

11. Taylor expanded in cosTheta_i around inf 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. +-commutative 99.8%

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

   4. exp-sum 99.8%

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

   5. add-exp-log 99.8%

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

   6. associate--r+ 99.8%

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

   7. associate-*r/ 99.8%

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

   8. associate-*r/ 99.8%

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

   9. sub-div 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in cosTheta_i around inf 99.8%

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

8. Taylor expanded in cosTheta_O around 0 99.8%

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

   1. distribute-neg-frac 99.8%

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

   2. metadata-eval 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 51.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{\frac{1}{\frac{\frac{v}{sinTheta\_i}}{sinTheta\_O}}}\\ \mathbf{elif}\;sinTheta\_i \cdot sinTheta\_O \leq 5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -2.0000000063421537e-30)
   (exp (/ 1.0 (/ (/ v sinTheta_i) sinTheta_O)))
   (if (<= (* sinTheta_i sinTheta_O) 5.000000179695649e-37)
     (/ 1.0 (/ v (* sinTheta_i sinTheta_O)))
     (exp (* sinTheta_O (/ sinTheta_i (- v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= -2.0000000063421537e-30f) {
		tmp = expf((1.0f / ((v / sinTheta_i) / sinTheta_O)));
	} else if ((sinTheta_i * sinTheta_O) <= 5.000000179695649e-37f) {
		tmp = 1.0f / (v / (sinTheta_i * sinTheta_O));
	} else {
		tmp = expf((sinTheta_O * (sinTheta_i / -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_i * sintheta_o) <= (-2.0000000063421537e-30)) then
        tmp = exp((1.0e0 / ((v / sintheta_i) / sintheta_o)))
    else if ((sintheta_i * sintheta_o) <= 5.000000179695649e-37) then
        tmp = 1.0e0 / (v / (sintheta_i * sintheta_o))
    else
        tmp = exp((sintheta_o * (sintheta_i / -v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(-2.0000000063421537e-30))
		tmp = exp(Float32(Float32(1.0) / Float32(Float32(v / sinTheta_i) / sinTheta_O)));
	elseif (Float32(sinTheta_i * sinTheta_O) <= Float32(5.000000179695649e-37))
		tmp = Float32(Float32(1.0) / Float32(v / Float32(sinTheta_i * sinTheta_O)));
	else
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / Float32(-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_i * sinTheta_O) <= single(-2.0000000063421537e-30))
		tmp = exp((single(1.0) / ((v / sinTheta_i) / sinTheta_O)));
	elseif ((sinTheta_i * sinTheta_O) <= single(5.000000179695649e-37))
		tmp = single(1.0) / (v / (sinTheta_i * sinTheta_O));
	else
		tmp = exp((sinTheta_O * (sinTheta_i / -v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -2e-30

   1. Initial program 99.3%

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

      1. associate-+l+ 99.3%

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

      2. associate--l- 99.3%

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

      3. associate-/l* 99.3%

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

      4. associate-/l* 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in sinTheta_i around inf 5.0%

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

      1. associate-*r/ 5.0%

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

      2. mul-1-neg 5.0%

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

      3. distribute-lft-neg-out 5.0%

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

      4. *-commutative 5.0%

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

   7. Simplified 5.0%

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

      1. clear-num 5.0%

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

      2. inv-pow 5.0%

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

      3. add-sqr-sqrt 2.7%

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

      4. sqrt-unprod 27.6%

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

      5. sqr-neg 27.6%

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

      6. sqrt-unprod 27.0%

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

      7. add-sqr-sqrt 50.0%

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

   9. Applied egg-rr 50.0%

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

       1. unpow-1 50.0%

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

       2. associate-/r* 50.0%

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

   11. Simplified 50.0%

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

   if -2e-30 < (*.f32 sinTheta_i sinTheta_O) < 5.00000018e-37

   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. associate-+l+ 99.8%

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

      2. associate--l- 99.8%

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

      3. associate-/l* 99.8%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 99.8%

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

      6. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in sinTheta_i around inf 6.4%

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

      1. associate-*r/ 6.4%

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

      2. mul-1-neg 6.4%

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

      3. distribute-lft-neg-out 6.4%

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

      4. *-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in sinTheta_i around 0 6.4%

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

      1. mul-1-neg 6.4%

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

      2. *-commutative 6.4%

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

      3. associate-*r/ 6.4%

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

      4. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

      5. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

      6. *-commutative 6.4%

         \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

      7. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   10. Simplified 6.4%

       \[\leadsto \color{blue}{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   11. Taylor expanded in sinTheta_O around inf 57.1%

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

       1. mul-1-neg 57.1%

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

       2. distribute-frac-neg2 57.1%

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

       3. associate-*r/ 26.8%

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

   13. Simplified 26.8%

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

       1. associate-*r/ 57.1%

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

       2. *-commutative 57.1%

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

       3. clear-num 57.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{-v}{sinTheta\_i \cdot sinTheta\_O}}} \]

       4. add-sqr-sqrt -0.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

       5. sqrt-unprod 34.9%

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

       6. sqr-neg 34.9%

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{v \cdot v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

       7. sqrt-unprod 57.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

       8. add-sqr-sqrt 57.9%

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

       9. *-commutative 57.9%

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

   15. Applied egg-rr 57.9%

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

   if 5.00000018e-37 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate--l- 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in sinTheta_i around inf 45.8%

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

      1. mul-1-neg 45.8%

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

      2. associate-/l* 45.8%

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

      3. distribute-rgt-neg-in 45.8%

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

      4. distribute-frac-neg2 45.8%

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

   7. Simplified 45.8%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{\frac{1}{\frac{\frac{v}{sinTheta\_i}}{sinTheta\_O}}}\\ \mathbf{elif}\;sinTheta\_i \cdot sinTheta\_O \leq 5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\\ \mathbf{elif}\;sinTheta\_i \cdot sinTheta\_O \leq 5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -2.0000000063421537e-30)
   (exp (* sinTheta_O (/ sinTheta_i v)))
   (if (<= (* sinTheta_i sinTheta_O) 5.000000179695649e-37)
     (/ 1.0 (/ v (* sinTheta_i sinTheta_O)))
     (exp (* sinTheta_O (/ sinTheta_i (- v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= -2.0000000063421537e-30f) {
		tmp = expf((sinTheta_O * (sinTheta_i / v)));
	} else if ((sinTheta_i * sinTheta_O) <= 5.000000179695649e-37f) {
		tmp = 1.0f / (v / (sinTheta_i * sinTheta_O));
	} else {
		tmp = expf((sinTheta_O * (sinTheta_i / -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_i * sintheta_o) <= (-2.0000000063421537e-30)) then
        tmp = exp((sintheta_o * (sintheta_i / v)))
    else if ((sintheta_i * sintheta_o) <= 5.000000179695649e-37) then
        tmp = 1.0e0 / (v / (sintheta_i * sintheta_o))
    else
        tmp = exp((sintheta_o * (sintheta_i / -v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(-2.0000000063421537e-30))
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / v)));
	elseif (Float32(sinTheta_i * sinTheta_O) <= Float32(5.000000179695649e-37))
		tmp = Float32(Float32(1.0) / Float32(v / Float32(sinTheta_i * sinTheta_O)));
	else
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / Float32(-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_i * sinTheta_O) <= single(-2.0000000063421537e-30))
		tmp = exp((sinTheta_O * (sinTheta_i / v)));
	elseif ((sinTheta_i * sinTheta_O) <= single(5.000000179695649e-37))
		tmp = single(1.0) / (v / (sinTheta_i * sinTheta_O));
	else
		tmp = exp((sinTheta_O * (sinTheta_i / -v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -2e-30

   1. Initial program 99.3%

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

      1. associate-+l+ 99.3%

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

      2. associate--l- 99.3%

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

      3. associate-/l* 99.3%

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

      4. associate-/l* 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in sinTheta_i around inf 5.0%

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

      1. associate-*r/ 5.0%

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

      2. mul-1-neg 5.0%

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

      3. distribute-lft-neg-out 5.0%

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

      4. *-commutative 5.0%

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

   7. Simplified 5.0%

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

      1. add-log-exp 5.0%

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

      2. *-un-lft-identity 5.0%

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

      3. log-prod 5.0%

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

      4. metadata-eval 5.0%

         \[\leadsto e^{\color{blue}{0} + \log \left(e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}\right)} \]

      5. add-log-exp 5.0%

         \[\leadsto e^{0 + \color{blue}{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}} \]

      6. associate-/l* 5.0%

         \[\leadsto e^{0 + \color{blue}{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}}} \]

      7. add-sqr-sqrt 2.7%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{-sinTheta\_O} \cdot \sqrt{-sinTheta\_O}}}{v}} \]

      8. sqrt-unprod 27.6%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{\left(-sinTheta\_O\right) \cdot \left(-sinTheta\_O\right)}}}{v}} \]

      9. sqr-neg 27.6%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

      10. sqrt-unprod 27.0%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{v}} \]

      11. add-sqr-sqrt 50.0%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{sinTheta\_O}}{v}} \]

   9. Applied egg-rr 50.0%

      \[\leadsto e^{\color{blue}{0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}}} \]
   10. Step-by-step derivation

       1. +-lft-identity 50.0%

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

       2. associate-*r/ 50.0%

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

       3. *-commutative 50.0%

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

       4. associate-*r/ 50.0%

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

   11. Simplified 50.0%

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

   if -2e-30 < (*.f32 sinTheta_i sinTheta_O) < 5.00000018e-37

   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. associate-+l+ 99.8%

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

      2. associate--l- 99.8%

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

      3. associate-/l* 99.8%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 99.8%

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

      6. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in sinTheta_i around inf 6.4%

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

      1. associate-*r/ 6.4%

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

      2. mul-1-neg 6.4%

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

      3. distribute-lft-neg-out 6.4%

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

      4. *-commutative 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in sinTheta_i around 0 6.4%

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

      1. mul-1-neg 6.4%

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

      2. *-commutative 6.4%

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

      3. associate-*r/ 6.4%

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

      4. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

      5. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

      6. *-commutative 6.4%

         \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

      7. associate-*r/ 6.4%

         \[\leadsto 1 - \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   10. Simplified 6.4%

       \[\leadsto \color{blue}{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   11. Taylor expanded in sinTheta_O around inf 57.1%

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

       1. mul-1-neg 57.1%

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

       2. distribute-frac-neg2 57.1%

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

       3. associate-*r/ 26.8%

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

   13. Simplified 26.8%

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

       1. associate-*r/ 57.1%

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

       2. *-commutative 57.1%

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

       3. clear-num 57.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{-v}{sinTheta\_i \cdot sinTheta\_O}}} \]

       4. add-sqr-sqrt -0.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

       5. sqrt-unprod 34.9%

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

       6. sqr-neg 34.9%

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{v \cdot v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

       7. sqrt-unprod 57.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

       8. add-sqr-sqrt 57.9%

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

       9. *-commutative 57.9%

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

   15. Applied egg-rr 57.9%

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

   if 5.00000018e-37 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate--l- 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in sinTheta_i around inf 45.8%

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

      1. mul-1-neg 45.8%

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

      2. associate-/l* 45.8%

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

      3. distribute-rgt-neg-in 45.8%

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

      4. distribute-frac-neg2 45.8%

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

   7. Simplified 45.8%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\\ \mathbf{elif}\;sinTheta\_i \cdot sinTheta\_O \leq 5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -2.0000000063421537e-30)
   (exp (* sinTheta_O (/ sinTheta_i v)))
   (/ 1.0 (/ v (* sinTheta_i sinTheta_O)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= -2.0000000063421537e-30f) {
		tmp = expf((sinTheta_O * (sinTheta_i / v)));
	} else {
		tmp = 1.0f / (v / (sinTheta_i * sinTheta_O));
	}
	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_i * sintheta_o) <= (-2.0000000063421537e-30)) then
        tmp = exp((sintheta_o * (sintheta_i / v)))
    else
        tmp = 1.0e0 / (v / (sintheta_i * sintheta_o))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(-2.0000000063421537e-30))
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / v)));
	else
		tmp = Float32(Float32(1.0) / Float32(v / Float32(sinTheta_i * sinTheta_O)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(-2.0000000063421537e-30))
		tmp = exp((sinTheta_O * (sinTheta_i / v)));
	else
		tmp = single(1.0) / (v / (sinTheta_i * sinTheta_O));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -2e-30

   1. Initial program 99.3%

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

      1. associate-+l+ 99.3%

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

      2. associate--l- 99.3%

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

      3. associate-/l* 99.3%

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

      4. associate-/l* 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in sinTheta_i around inf 5.0%

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

      1. associate-*r/ 5.0%

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

      2. mul-1-neg 5.0%

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

      3. distribute-lft-neg-out 5.0%

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

      4. *-commutative 5.0%

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

   7. Simplified 5.0%

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

      1. add-log-exp 5.0%

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

      2. *-un-lft-identity 5.0%

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

      3. log-prod 5.0%

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

      4. metadata-eval 5.0%

         \[\leadsto e^{\color{blue}{0} + \log \left(e^{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}\right)} \]

      5. add-log-exp 5.0%

         \[\leadsto e^{0 + \color{blue}{\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}}} \]

      6. associate-/l* 5.0%

         \[\leadsto e^{0 + \color{blue}{sinTheta\_i \cdot \frac{-sinTheta\_O}{v}}} \]

      7. add-sqr-sqrt 2.7%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{-sinTheta\_O} \cdot \sqrt{-sinTheta\_O}}}{v}} \]

      8. sqrt-unprod 27.6%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{\left(-sinTheta\_O\right) \cdot \left(-sinTheta\_O\right)}}}{v}} \]

      9. sqr-neg 27.6%

         \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v}} \]

      10. sqrt-unprod 27.0%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{v}} \]

      11. add-sqr-sqrt 50.0%

          \[\leadsto e^{0 + sinTheta\_i \cdot \frac{\color{blue}{sinTheta\_O}}{v}} \]

   9. Applied egg-rr 50.0%

      \[\leadsto e^{\color{blue}{0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}}} \]
   10. Step-by-step derivation

       1. +-lft-identity 50.0%

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

       2. associate-*r/ 50.0%

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

       3. *-commutative 50.0%

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

       4. associate-*r/ 50.0%

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

   11. Simplified 50.0%

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

   if -2e-30 < (*.f32 sinTheta_i sinTheta_O)

   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. associate-+l+ 99.8%

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

      2. associate--l- 99.8%

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

      3. associate-/l* 99.8%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 99.8%

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

      6. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in sinTheta_i around inf 16.6%

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

      1. associate-*r/ 16.6%

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

      2. mul-1-neg 16.6%

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

      3. distribute-lft-neg-out 16.6%

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

      4. *-commutative 16.6%

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

   7. Simplified 16.6%

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

   8. Taylor expanded in sinTheta_i around 0 6.3%

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

      1. mul-1-neg 6.3%

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

      2. *-commutative 6.3%

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

      3. associate-*r/ 6.3%

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

      4. unsub-neg 6.3%

         \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

      5. associate-*r/ 6.3%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

      6. *-commutative 6.3%

         \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

      7. associate-*r/ 6.3%

         \[\leadsto 1 - \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   10. Simplified 6.3%

       \[\leadsto \color{blue}{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

   11. Taylor expanded in sinTheta_O around inf 44.2%

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

       1. mul-1-neg 44.2%

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

       2. distribute-frac-neg2 44.2%

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

       3. associate-*r/ 21.7%

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

   13. Simplified 21.7%

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

       1. associate-*r/ 44.2%

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

       2. *-commutative 44.2%

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

       3. clear-num 44.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{-v}{sinTheta\_i \cdot sinTheta\_O}}} \]

       4. add-sqr-sqrt -0.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

       5. sqrt-unprod 27.5%

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

       6. sqr-neg 27.5%

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{v \cdot v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

       7. sqrt-unprod 44.8%

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

       8. add-sqr-sqrt 44.8%

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

       9. *-commutative 44.8%

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

   15. Applied egg-rr 44.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% 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 99.7%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. associate--l- 99.7%

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

   7. associate-/l* 99.7%

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

   8. associate-/l* 99.7%

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

   9. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in cosTheta_O around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 97.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in v around 0 97.5%

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

7. Final simplification 97.5%

   \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i + -1}{v}} \]
8. Add Preprocessing

Alternative 8: 39.2% accurate, 31.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (v / (sinTheta_i * sinTheta_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 = 1.0e0 / (v / (sintheta_i * sintheta_o))
end function

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 14.6%

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

   1. associate-*r/ 14.6%

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

   2. mul-1-neg 14.6%

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

   3. distribute-lft-neg-out 14.6%

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

   4. *-commutative 14.6%

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

7. Simplified 14.6%

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

8. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. mul-1-neg 6.3%

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

   2. *-commutative 6.3%

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

   3. associate-*r/ 6.3%

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

   4. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

   5. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

   6. *-commutative 6.3%

      \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

   7. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

10. Simplified 6.3%

    \[\leadsto \color{blue}{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 37.8%

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

    1. mul-1-neg 37.8%

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

    2. distribute-frac-neg2 37.8%

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

    3. associate-*r/ 19.1%

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

13. Simplified 19.1%

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

    1. associate-*r/ 37.8%

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

    2. *-commutative 37.8%

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

    3. clear-num 38.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{-v}{sinTheta\_i \cdot sinTheta\_O}}} \]

    4. add-sqr-sqrt -0.0%

       \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

    5. sqrt-unprod 23.8%

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

    6. sqr-neg 23.8%

       \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{v \cdot v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

    7. sqrt-unprod 38.4%

       \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{sinTheta\_i \cdot sinTheta\_O}} \]

    8. add-sqr-sqrt 38.4%

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

    9. *-commutative 38.4%

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

15. Applied egg-rr 38.4%

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

16. Final simplification 38.4%

    \[\leadsto \frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}} \]
17. Add Preprocessing

Alternative 9: 38.8% accurate, 37.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (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 = (sintheta_i * sintheta_o) / -v
end function

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 14.6%

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

   1. associate-*r/ 14.6%

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

   2. mul-1-neg 14.6%

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

   3. distribute-lft-neg-out 14.6%

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

   4. *-commutative 14.6%

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

7. Simplified 14.6%

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

8. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. mul-1-neg 6.3%

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

   2. *-commutative 6.3%

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

   3. associate-*r/ 6.3%

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

   4. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

   5. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

   6. *-commutative 6.3%

      \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

   7. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

10. Simplified 6.3%

    \[\leadsto \color{blue}{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 37.8%

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

    1. mul-1-neg 37.8%

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

    2. distribute-frac-neg2 37.8%

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

    3. associate-*r/ 19.1%

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

13. Simplified 19.1%

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

14. Taylor expanded in sinTheta_O around 0 37.8%

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

    1. mul-1-neg 37.8%

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

    2. distribute-frac-neg 37.8%

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

    3. distribute-rgt-neg-in 37.8%

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

16. Simplified 37.8%

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

17. Final simplification 37.8%

    \[\leadsto \frac{sinTheta\_i \cdot sinTheta\_O}{-v} \]
18. Add Preprocessing

Alternative 10: 20.8% accurate, 37.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 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 = sintheta_i * (sintheta_o / -v)
end function

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 14.6%

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

   1. associate-*r/ 14.6%

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

   2. mul-1-neg 14.6%

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

   3. distribute-lft-neg-out 14.6%

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

   4. *-commutative 14.6%

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

7. Simplified 14.6%

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

8. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. mul-1-neg 6.3%

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

   2. *-commutative 6.3%

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

   3. associate-*r/ 6.3%

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

   4. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

   5. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

   6. *-commutative 6.3%

      \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

   7. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

10. Simplified 6.3%

    \[\leadsto \color{blue}{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 37.8%

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

    1. mul-1-neg 37.8%

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

    2. distribute-frac-neg2 37.8%

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

    3. associate-*r/ 19.1%

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

13. Simplified 19.1%

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

14. Taylor expanded in sinTheta_O around 0 37.8%

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

    1. associate-*l/ 19.1%

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

    2. associate-*r* 19.1%

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

    3. *-commutative 19.1%

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

    4. mul-1-neg 19.1%

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

    5. distribute-neg-frac2 19.1%

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

16. Simplified 19.1%

    \[\leadsto \color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \]
17. Add Preprocessing

Alternative 11: 20.8% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 14.6%

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

   1. associate-*r/ 14.6%

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

   2. mul-1-neg 14.6%

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

   3. distribute-lft-neg-out 14.6%

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

   4. *-commutative 14.6%

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

7. Simplified 14.6%

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

8. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. mul-1-neg 6.3%

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

   2. *-commutative 6.3%

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

   3. associate-*r/ 6.3%

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

   4. unsub-neg 6.3%

      \[\leadsto \color{blue}{1 - sinTheta\_i \cdot \frac{sinTheta\_O}{v}} \]

   5. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \]

   6. *-commutative 6.3%

      \[\leadsto 1 - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} \]

   7. associate-*r/ 6.3%

      \[\leadsto 1 - \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

10. Simplified 6.3%

    \[\leadsto \color{blue}{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \]

11. Taylor expanded in sinTheta_O around inf 37.8%

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

    1. mul-1-neg 37.8%

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

    2. distribute-frac-neg2 37.8%

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

    3. associate-*r/ 19.1%

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

13. Simplified 19.1%

    \[\leadsto \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}} \]
14. 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 99.7%

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

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 14.6%

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

   1. associate-*r/ 14.6%

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

   2. mul-1-neg 14.6%

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

   3. distribute-lft-neg-out 14.6%

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

   4. *-commutative 14.6%

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

7. Simplified 14.6%

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

8. Taylor expanded in sinTheta_i around 0 6.3%

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

   1. mul-1-neg 6.3%

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

   2. *-commutative 6.3%

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

   3. associate-*r/ 6.3%

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

   4. distribute-rgt-neg-in 6.3%

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

   5. distribute-frac-neg 6.3%

      \[\leadsto 1 + sinTheta\_i \cdot \color{blue}{\frac{-sinTheta\_O}{v}} \]

10. Simplified 6.3%

    \[\leadsto \color{blue}{1 + sinTheta\_i \cdot \frac{-sinTheta\_O}{v}} \]

11. Taylor expanded in sinTheta_i around 0 6.4%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024102 
(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))))))