HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.6%

Time: 10.3s

Alternatives: 8

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(0.6931e0) * (0.5e0 / v)) * (1.0e0 / exp((((-1.0e0) - (sintheta_i * sintheta_o)) / -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)) * Float32(Float32(1.0) / exp(Float32(Float32(Float32(-1.0) - Float32(sinTheta_i * sinTheta_O)) / Float32(-v)))))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]

   4. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   5. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)\right) + \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right)}} \]

   6. exp-sum N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}} \]

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around 0

   \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1 \cdot \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}}{v}} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1 \cdot 1 + -1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}}{v}} \]

   2. metadata-eval N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1} + -1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}} \]

   3. mul-1-neg N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{-1 + \color{blue}{\left(\mathsf{neg}\left(sinTheta\_O \cdot sinTheta\_i\right)\right)}}{v}} \]

   4. unsub-neg N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   5. lower--.f32 N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   6. *-commutative N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{-1 - \color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}} \]

   7. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot \color{blue}{e^{\frac{-1 - sinTheta\_i \cdot sinTheta\_O}{v}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\color{blue}{\frac{-1 - sinTheta\_i \cdot sinTheta\_O}{v}}} \]

   3. frac-2neg N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\color{blue}{\frac{\mathsf{neg}\left(\left(-1 - sinTheta\_i \cdot sinTheta\_O\right)\right)}{\mathsf{neg}\left(v\right)}}} \]

   4. distribute-frac-neg2 N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left(-1 - sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right)}} \]

   5. exp-neg N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot \color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\left(-1 - sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}}}} \]

   6. lower-/.f32 N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot \color{blue}{\frac{1}{e^{\frac{\mathsf{neg}\left(\left(-1 - sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}}}} \]

   7. lower-exp.f32 N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot \frac{1}{\color{blue}{e^{\frac{\mathsf{neg}\left(\left(-1 - sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}}}} \]

   8. lower-/.f32 N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(\left(-1 - sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}}}} \]

   9. lower-neg.f32 99.7

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(0.6931e0) * (0.5e0 / v)) * exp((((-1.0e0) - (sintheta_i * sintheta_o)) / 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(Float32(-1.0) - Float32(sinTheta_i * sinTheta_O)) / v)))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]

   4. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   5. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)\right) + \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right)}} \]

   6. exp-sum N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}}} \]

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around 0

   \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1 \cdot \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}}{v}} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1 \cdot 1 + -1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}}{v}} \]

   2. metadata-eval N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1} + -1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}} \]

   3. mul-1-neg N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{-1 + \color{blue}{\left(\mathsf{neg}\left(sinTheta\_O \cdot sinTheta\_i\right)\right)}}{v}} \]

   4. unsub-neg N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   5. lower--.f32 N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   6. *-commutative N/A

      \[\leadsto \left(e^{\frac{6931}{10000}} \cdot \frac{\frac{1}{2}}{v}\right) \cdot e^{\frac{-1 - \color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}} \]

   7. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 2.1× 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 99.6%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   2. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   5. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   8. sub-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}}} \]

5. Applied rewrites 13.8%

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

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{1}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 99.6%

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

Alternative 4: 17.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 4.999999898305949e-32)
   (exp (* cosTheta_O (/ cosTheta_i v)))
   (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) <= 4.999999898305949e-32f) {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	} 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) <= 4.999999898305949e-32) then
        tmp = exp((costheta_o * (costheta_i / v)))
    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(4.999999898305949e-32))
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	else
		tmp = exp(Float32(Float32(-sinTheta_O) * Float32(sinTheta_i / 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(4.999999898305949e-32))
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	else
		tmp = exp((-sinTheta_O * (sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 4.9999999e-32

   1. Initial program 99.5%

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

      1. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-log.f32 N/A

         \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. lift-/.f32 N/A

         \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

      4. log-rec N/A

         \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

      5. unsub-neg N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)}} \]

      6. lower--.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)}} \]

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 12.2

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

   7. Applied rewrites 12.2%

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

   9. Applied rewrites 12.2%

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

   if 4.9999999e-32 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 100.0%

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

      1. lift-+.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

      2. lift-log.f32 N/A

         \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

      3. lift-/.f32 N/A

         \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

      4. log-rec N/A

         \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

      5. unsub-neg N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)}} \]

      6. lower--.f32 N/A

         \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)}} \]

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 8.4

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

   7. Applied rewrites 8.4%

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

   8. Taylor expanded in sinTheta_i around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-/.f32 48.7

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

   10. Applied rewrites 48.7%

       \[\leadsto e^{\color{blue}{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.4% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 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 (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 = (0.5e0 / v) * exp(((-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(-1.0) / v)))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   2. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   5. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   8. sub-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}}} \]

5. Applied rewrites 15.7%

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

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{1}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 99.6%

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

9. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{-1}{v}} \]
10. Step-by-step derivation

11. Applied rewrites 96.6%

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

Alternative 6: 98.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-log.f32 N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. lift-/.f32 N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

   4. log-rec N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

   5. unsub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)}} \]

   6. lower--.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)}} \]

4. Applied rewrites 99.6%

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

5. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\left(\frac{6931}{10000} + \frac{\color{blue}{-1 \cdot \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}}{v}\right) - \log \left(2 \cdot v\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto e^{\left(\frac{6931}{10000} + \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}}{v}\right) - \log \left(2 \cdot v\right)} \]

   2. metadata-eval N/A

      \[\leadsto e^{\left(\frac{6931}{10000} + \frac{\color{blue}{-1} + -1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right) - \log \left(2 \cdot v\right)} \]

   3. mul-1-neg N/A

      \[\leadsto e^{\left(\frac{6931}{10000} + \frac{-1 + \color{blue}{\left(\mathsf{neg}\left(sinTheta\_O \cdot sinTheta\_i\right)\right)}}{v}\right) - \log \left(2 \cdot v\right)} \]

   4. unsub-neg N/A

      \[\leadsto e^{\left(\frac{6931}{10000} + \frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v}\right) - \log \left(2 \cdot v\right)} \]

   5. lower--.f32 N/A

      \[\leadsto e^{\left(\frac{6931}{10000} + \frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v}\right) - \log \left(2 \cdot v\right)} \]

   6. *-commutative N/A

      \[\leadsto e^{\left(\frac{6931}{10000} + \frac{-1 - \color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}\right) - \log \left(2 \cdot v\right)} \]

   7. lower-*.f32 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. associate--r+ N/A

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

   3. lower--.f32 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 94.8

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

10. Applied rewrites 94.8%

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

Alternative 7: 12.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-log.f32 N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. lift-/.f32 N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}} \]

   4. log-rec N/A

      \[\leadsto e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \color{blue}{\left(\mathsf{neg}\left(\log \left(2 \cdot v\right)\right)\right)}} \]

   5. unsub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)}} \]

   6. lower--.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)}} \]

4. Applied rewrites 99.6%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 11.6

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

7. Applied rewrites 11.6%

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

9. Applied rewrites 11.6%

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

Alternative 8: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. +-commutative N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   4. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

   5. lift-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   6. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

   8. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   10. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   11. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \]

   13. lower-exp.f32 99.6

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

   14. lift-+.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}} \]

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{v} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{2 \cdot v}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   5. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot v}{1}}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   6. inv-pow N/A

      \[\leadsto \color{blue}{{\left(\frac{2 \cdot v}{1}\right)}^{-1}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   7. sqr-pow N/A

      \[\leadsto \color{blue}{\left({\left(\frac{2 \cdot v}{1}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{2 \cdot v}{1}\right)}^{\left(\frac{-1}{2}\right)}\right)} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   8. pow2 N/A

      \[\leadsto \color{blue}{{\left({\left(\frac{2 \cdot v}{1}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   9. lower-pow.f32 N/A

      \[\leadsto \color{blue}{{\left({\left(\frac{2 \cdot v}{1}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   10. metadata-eval N/A

       \[\leadsto {\left({\left(\frac{2 \cdot v}{1}\right)}^{\color{blue}{\frac{-1}{2}}}\right)}^{2} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   11. lower-pow.f32 N/A

       \[\leadsto {\color{blue}{\left({\left(\frac{2 \cdot v}{1}\right)}^{\frac{-1}{2}}\right)}}^{2} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   12. lift-*.f32 N/A

       \[\leadsto {\left({\left(\frac{\color{blue}{2 \cdot v}}{1}\right)}^{\frac{-1}{2}}\right)}^{2} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   13. associate-/l* N/A

       \[\leadsto {\left({\color{blue}{\left(2 \cdot \frac{v}{1}\right)}}^{\frac{-1}{2}}\right)}^{2} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   14. /-rgt-identity N/A

       \[\leadsto {\left({\left(2 \cdot \color{blue}{v}\right)}^{\frac{-1}{2}}\right)}^{2} \cdot e^{\frac{6931}{10000} + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

   15. lift-*.f32 99.6

       \[\leadsto {\left({\color{blue}{\left(2 \cdot v\right)}}^{-0.5}\right)}^{2} \cdot e^{0.6931 + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{{\left({\left(2 \cdot v\right)}^{-0.5}\right)}^{2}} \cdot e^{0.6931 + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}} \]

7. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}} \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2}}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{\frac{1}{2}}\right)}^{2} \cdot e^{\frac{6931}{10000}}}}{v} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}\right)} \cdot e^{\frac{6931}{10000}}}{v} \]

   3. rem-square-sqrt N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2}} \cdot e^{\frac{6931}{10000}}}{v} \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000}}}{v}} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{v} \cdot \frac{1}{2}} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{v} \cdot \frac{1}{2}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000}}}{v}} \cdot \frac{1}{2} \]

   8. lower-exp.f32 4.7

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

9. Applied rewrites 4.7%

   \[\leadsto \color{blue}{\frac{e^{0.6931}}{v} \cdot 0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024299 
(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))))))