HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 12.3s

Alternatives: 10

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((((single(-1.0) / v) + single(0.6931)) - log((single(2.0) * 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. 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 99.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in sinTheta_i around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 99.3

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

10. Applied rewrites 99.3%

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

11. Taylor expanded in cosTheta_O around 0

    \[\leadsto e^{\left(\frac{6931}{10000} + \frac{-1}{v}\right) - \log \left(2 \cdot v\right)} \]
12. Step-by-step derivation

13. Applied rewrites 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Taylor expanded in cosTheta_O 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 16.4%

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

6. Taylor expanded in sinTheta_O around -inf

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

8. Applied rewrites 99.6%

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

10. Applied rewrites 99.6%

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

11. Taylor expanded in sinTheta_i around 0

    \[\leadsto \frac{1}{2} \cdot e^{\frac{6931}{10000} - \left(\log v + \frac{1}{v}\right)} \]
12. Step-by-step derivation

13. Applied rewrites 99.6%

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

14. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

3. Taylor expanded in cosTheta_O 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 16.4%

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

6. Taylor expanded in sinTheta_O around -inf

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 97.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -5.002635517639597 \cdot 10^{-43}:\\ \;\;\;\;e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{fma}\left(-sinTheta\_O, sinTheta\_i, -1\right)}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* cosTheta_i cosTheta_O) -5.002635517639597e-43)
   (exp (/ (fma cosTheta_i cosTheta_O -1.0) v))
   (exp (/ (fma (- sinTheta_O) sinTheta_i -1.0) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -5.002635517639597e-43f) {
		tmp = expf((fmaf(cosTheta_i, cosTheta_O, -1.0f) / v));
	} else {
		tmp = expf((fmaf(-sinTheta_O, sinTheta_i, -1.0f) / v));
	}
	return tmp;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-5.002635517639597e-43))
		tmp = exp(Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v));
	else
		tmp = exp(Float32(fma(Float32(-sinTheta_O), sinTheta_i, Float32(-1.0)) / v));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -5.00264e-43

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing
   3. 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 99.7%

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

   5. Taylor expanded in v around 0

      \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
   6. 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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. associate--l+ N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f32 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f32 N/A

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

      16. sub-neg N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f32 89.7

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in sinTheta_i around 0

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

   10. Applied rewrites 97.0%

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

   if -5.00264e-43 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.7%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing
   3. 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 99.7%

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

   5. Taylor expanded in v around 0

      \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
   6. 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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. associate--l+ N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f32 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f32 N/A

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

      16. sub-neg N/A

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

      17. *-commutative N/A

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

      18. metadata-eval N/A

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

      19. lower-fma.f32 93.2

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

   7. Applied rewrites 93.2%

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

   8. Taylor expanded in cosTheta_O around 0

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

   10. Applied rewrites 95.8%

       \[\leadsto e^{\frac{\mathsf{fma}\left(-sinTheta\_O, sinTheta\_i, -1\right)}{v}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

3. Taylor expanded in cosTheta_O 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 16.4%

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

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. Final simplification 99.6%

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

Alternative 6: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(-1.0) / v)) * (single(0.5) / 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. Add Preprocessing

3. Taylor expanded in cosTheta_O 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 16.4%

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

6. Taylor expanded in sinTheta_O around -inf

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in sinTheta_i around 0

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

11. Applied rewrites 99.1%

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

12. Taylor expanded in v around 0

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

14. Applied rewrites 97.2%

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

15. Final simplification 97.2%

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

Alternative 7: 97.8% accurate, 2.3× speedup

Math

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

FPCore

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(fma(cosTheta_i, cosTheta_O, Float32(-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. 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 99.7%

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

5. Taylor expanded in v around 0

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
6. 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. sub-neg N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. metadata-eval N/A

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

   7. associate-+l+ N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. associate--l+ N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f32 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f32 N/A

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

   16. sub-neg N/A

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

   17. *-commutative N/A

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

   18. metadata-eval N/A

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

   19. lower-fma.f32 91.0

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

7. Applied rewrites 91.0%

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

8. Taylor expanded in sinTheta_i around 0

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

10. Applied rewrites 95.8%

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

Alternative 8: 13.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((cosTheta_O / v) * cosTheta_i));
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. 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 99.7%

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

5. Taylor expanded in cosTheta_O 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 12.9

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

7. Applied rewrites 12.9%

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

9. Applied rewrites 12.9%

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

Alternative 9: 13.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((cosTheta_i / v) * cosTheta_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. 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 99.7%

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

5. Taylor expanded in cosTheta_O 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 12.9

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

7. Applied rewrites 12.9%

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

9. Applied rewrites 12.9%

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

10. Final simplification 12.9%

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

Alternative 10: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

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)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(single(0.6931)) * (single(0.5) / 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. Add Preprocessing

3. Taylor expanded in cosTheta_O 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 16.4%

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

6. Taylor expanded in v around inf

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

8. Applied rewrites 4.7%

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

9. Final simplification 4.7%

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

Reproduce

herbie shell --seed 2024248 
(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))))))