HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.8%

Time: 4.3s

Alternatives: 8

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

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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))))))

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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))))))

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)
use fmin_fmax_functions
    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: 99.8% accurate, 1.1× speedup

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

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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)) / exp(((-0.6931e0) + ((-1.0e0) / ((-1.0e0) / log((v + v))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in sinTheta_i around 0

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in cosTheta_i around 0

   \[\leadsto \frac{e^{\frac{-1}{v}}}{e^{-0.6931 + \log \left(v + v\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 99.8%

   \[\leadsto \frac{e^{\frac{-1}{v}}}{e^{-0.6931 + \log \left(v + v\right)}} \]
10. Step-by-step derivation

    1. *-lft-identity N/A

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

    2. metadata-eval N/A

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

    3. associate-/r/ N/A

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

    4. lift-/.f32 N/A

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

    5. lift-/.f32 99.8%

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

11. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

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

Math

\[0.999952795349722 \cdot \left(e^{-\log v} \cdot e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right) \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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)))
  (*
 0.999952795349722
 (* (exp (- (log v))) (exp (/ (- (* cosTheta_O cosTheta_i) 1.0) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.999952795349722f * (expf(-logf(v)) * expf((((cosTheta_O * cosTheta_i) - 1.0f) / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 = 0.999952795349722e0 * (exp(-log(v)) * exp((((costheta_o * costheta_i) - 1.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.999952795349722) * Float32(exp(Float32(-log(v))) * exp(Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(1.0)) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.999952795349722) * (exp(-log(v)) * exp((((cosTheta_O * cosTheta_i) - single(1.0)) / v)));
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \cdot \frac{1}{v}}{e^{-0.6931} \cdot 2} \]
4. Step-by-step derivation

5. Applied rewrites 99.6%

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

6. Evaluated real constant 99.6%

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \cdot e^{-\log v}}{1.000047206878662} \]

7. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{2097152}{2097251} \cdot \left(e^{\mathsf{neg}\left(\log v\right)} \cdot e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.6%

   \[\leadsto 0.999952795349722 \cdot \left(e^{-\log v} \cdot e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right) \]
10. Add Preprocessing

Alternative 3: 99.6% accurate, 2.3× speedup

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

Math

\[0.999952795349722 \cdot \frac{e^{\frac{-1}{v}}}{v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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)))
  (* 0.999952795349722 (/ (exp (/ -1.0 v)) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.999952795349722f * (expf((-1.0f / v)) / v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 = 0.999952795349722e0 * (exp(((-1.0e0) / v)) / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.999952795349722) * Float32(exp(Float32(Float32(-1.0) / v)) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.999952795349722) * (exp((single(-1.0) / v)) / v);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Evaluated real constant 99.6%

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \cdot \frac{1}{v}}{1.000047206878662} \]

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \frac{2097152}{2097251} \cdot \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}}{v} \]
6. Step-by-step derivation

7. Applied rewrites 99.8%

   \[\leadsto 0.999952795349722 \cdot \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}}{v} \]

8. Taylor expanded in cosTheta_i around 0

   \[\leadsto 0.999952795349722 \cdot \frac{e^{\frac{-1}{v}}}{v} \]
9. Step-by-step derivation

10. Applied rewrites 99.8%

    \[\leadsto 0.999952795349722 \cdot \frac{e^{\frac{-1}{v}}}{v} \]
11. Add Preprocessing

Alternative 4: 98.1% accurate, 2.5× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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_O cosTheta_i) 1.0) v)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in v around 0

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 98.1%

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

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 98.1%

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

Alternative 5: 19.0% accurate, 2.0× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -1.0000000031710769 \cdot 10^{-30}:\\ \;\;\;\;e^{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta\_O \cdot sinTheta\_i}{v}}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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)))
  (if (<= (* cosTheta_i cosTheta_O) -1.0000000031710769e-30)
  (exp (/ (* cosTheta_O cosTheta_i) v))
  (exp (/ (- (* sinTheta_O sinTheta_i)) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -1.0000000031710769e-30f) {
		tmp = expf(((cosTheta_O * cosTheta_i) / v));
	} else {
		tmp = expf((-(sinTheta_O * sinTheta_i) / v));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((costheta_i * costheta_o) <= (-1.0000000031710769e-30)) then
        tmp = exp(((costheta_o * costheta_i) / v))
    else
        tmp = exp((-(sintheta_o * sintheta_i) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-1.0000000031710769e-30))
		tmp = exp(Float32(Float32(cosTheta_O * cosTheta_i) / v));
	else
		tmp = exp(Float32(Float32(-Float32(sinTheta_O * sinTheta_i)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((cosTheta_i * cosTheta_O) <= single(-1.0000000031710769e-30))
		tmp = exp(((cosTheta_O * cosTheta_i) / v));
	else
		tmp = exp((-(sinTheta_O * sinTheta_i) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -1e-30

   1. Initial program 99.6%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in cosTheta_i around inf

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

   7. Applied rewrites 13.3%

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

   if -1e-30 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.6%

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

   2. Taylor expanded in v around 0

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in sinTheta_i around inf

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

   7. Applied rewrites 13.4%

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

   9. Applied rewrites 13.4%

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

Alternative 6: 13.3% accurate, 2.8× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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))))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in v around 0

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

7. Applied rewrites 13.3%

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

9. Applied rewrites 13.3%

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

Alternative 7: 4.6% accurate, 6.7× speedup

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

Math

\[-0.999952795349722 \cdot \frac{-1}{v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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)))
  (* -0.999952795349722 (/ -1.0 v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return -0.999952795349722f * (-1.0f / v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 = (-0.999952795349722e0) * ((-1.0e0) / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(-0.999952795349722) * Float32(Float32(-1.0) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(-0.999952795349722) * (single(-1.0) / v);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in v around inf

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

6. Applied rewrites 4.6%

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

7. Evaluated real constant 4.6%

   \[\leadsto \frac{0.5}{v \cdot 0.500023603439331} \]
8. Step-by-step derivation

9. Applied rewrites 4.6%

   \[\leadsto -0.999952795349722 \cdot \frac{-1}{v} \]
10. Add Preprocessing

Alternative 8: 4.6% accurate, 11.0× speedup

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

Math

\[\frac{0.999952795349722}{v} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :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)))
  (/ 0.999952795349722 v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.999952795349722f / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
use fmin_fmax_functions
    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 = 0.999952795349722e0 / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.999952795349722) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.999952795349722) / v;
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Evaluated real constant 99.6%

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}} \cdot \frac{1}{v}}{1.000047206878662} \]

5. Taylor expanded in v around inf

   \[\leadsto \frac{\frac{2097152}{2097251}}{v} \]
6. Step-by-step derivation

7. Applied rewrites 4.6%

   \[\leadsto \frac{0.999952795349722}{v} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2026086 
(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))))))