HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 12.0s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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 99.9%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in cosTheta_i around inf

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

   1. *-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 99.9

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

10. Applied rewrites 99.9%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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(((((costheta_i * costheta_o) - 1.0e0) / v) + 0.6931e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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 99.9%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in cosTheta_i around inf

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

   1. *-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 99.9

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

10. Applied rewrites 99.9%

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

    1. lift-exp.f32 N/A

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

    2. lift--.f32 N/A

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

    3. sub-neg N/A

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

    4. lift-log.f32 N/A

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

    5. log-rec N/A

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

    6. lift-*.f32 N/A

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

    7. +-commutative N/A

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

    8. exp-sum N/A

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

    9. rem-exp-log N/A

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

    10. lower-*.f32 N/A

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

12. Applied rewrites 99.9%

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

Alternative 3: 97.9% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ (- -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(((-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((((-1.0e0) - (sintheta_i * sintheta_o)) / v))
end function

Julia

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

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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 99.9%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 99.9

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

7. Applied rewrites 99.9%

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

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 98.1

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

10. Applied rewrites 98.1%

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

11. Taylor expanded in cosTheta_i around 0

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

13. Applied rewrites 98.8%

    \[\leadsto e^{\frac{-1 - sinTheta\_i \cdot sinTheta\_O}{v}} \]
14. Add Preprocessing

Alternative 4: 13.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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 99.9%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 99.9

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

7. Applied rewrites 99.9%

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

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. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-/.f32 11.7

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

10. Applied rewrites 11.7%

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

Alternative 5: 13.3% 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 99.9%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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 99.9%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in cosTheta_i around inf

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}} \]
9. 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 9.1

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

10. Applied rewrites 9.1%

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

12. Applied rewrites 9.1%

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

Alternative 6: 4.6% accurate, 2.3× speedup

Math

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

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(Float32(exp(Float32(0.6931)) * 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 99.9%

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

3. Taylor expanded in v around -inf

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

   1. associate-+r+ N/A

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

   2. exp-sum N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. rem-exp-log N/A

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

   6. lower-*.f32 N/A

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

   7. exp-sum N/A

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

   8. rem-exp-log N/A

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

   9. lower-*.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 4.6

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

5. Applied rewrites 4.6%

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

7. Applied rewrites 4.6%

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

Alternative 7: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in v around -inf

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

   1. associate-+r+ N/A

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

   2. exp-sum N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. rem-exp-log N/A

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

   6. lower-*.f32 N/A

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

   7. exp-sum N/A

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

   8. rem-exp-log N/A

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

   9. lower-*.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 4.6

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

5. Applied rewrites 4.6%

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

7. Applied rewrites 4.6%

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

9. Applied rewrites 4.6%

   \[\leadsto \frac{1}{\left(2 \cdot v\right) \cdot \color{blue}{e^{-0.6931}}} \]
10. Step-by-step derivation

11. Applied rewrites 4.6%

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

Reproduce

herbie shell --seed 2024340 
(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))))))