HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 16.8s

Alternatives: 12

Speedup: 2.0×

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, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((((cosTheta_i / (v / cosTheta_O)) - ((sinTheta_i / (v / sinTheta_O)) + (single(1.0) / v))) + (single(0.6931) + log((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. Step-by-step derivation

   1. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. *-commutative 99.7%

      \[\leadsto e^{\left(\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   6. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Final simplification 99.7%

   \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_i around 0 99.6%

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

   1. exp-diff 99.6%

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

   2. div-inv 99.6%

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

   3. div-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. div-inv 99.3%

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

   6. distribute-rgt-out 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \left(e^{0.6931} \cdot \frac{1}{e^{\frac{1}{v} \cdot \left(1 + sinTheta_i \cdot sinTheta_O\right)}}\right) \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_i around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. exp-diff 99.6%

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

   3. frac-times 99.6%

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

   4. *-commutative 99.6%

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

   5. div-inv 99.6%

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

   6. *-commutative 99.6%

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

   7. div-inv 99.3%

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

   8. distribute-rgt-out 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \frac{0.5 \cdot e^{0.6931}}{v \cdot e^{\frac{1}{v} \cdot \left(1 + sinTheta_i \cdot sinTheta_O\right)}} \]

Alternative 4: 99.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* 0.5 (exp (- 0.6931 (* (/ 1.0 v) (+ 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 (0.5f * expf((0.6931f - ((1.0f / v) * (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 = (0.5e0 * exp((0.6931e0 - ((1.0e0 / v) * (1.0e0 + (sintheta_i * sintheta_o)))))) / v
end function

Julia

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_i around 0 99.6%

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

   1. associate-*l/ 99.6%

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

   2. div-inv 99.6%

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

   3. *-commutative 99.6%

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

   4. div-inv 99.3%

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

   5. distribute-rgt-out 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (- 0.6931 (/ (+ 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 (0.5f / v) * expf((0.6931f - ((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 = (0.5e0 / v) * exp((0.6931e0 - ((1.0e0 + (sintheta_i * sintheta_o)) / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_i around 0 99.6%

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

   1. exp-diff 99.6%

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

   2. frac-2neg 99.6%

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

   3. div-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. div-inv 99.3%

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

   6. distribute-rgt-out 99.6%

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

6. Applied egg-rr 99.6%

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

   1. exp-prod 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in v around 0 99.6%

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

    1. associate-*r/ 99.6%

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

    2. *-commutative 99.6%

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

    3. times-frac 99.6%

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

    4. div-exp 99.6%

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

    5. *-commutative 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 6: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_i around 0 99.6%

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

5. Taylor expanded in sinTheta_O around 0 99.6%

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

6. Final simplification 99.6%

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

Alternative 7: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_i around 0 99.6%

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

   1. exp-diff 99.6%

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

   2. frac-2neg 99.6%

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

   3. div-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. div-inv 99.3%

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

   6. distribute-rgt-out 99.6%

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

6. Applied egg-rr 99.6%

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

   1. exp-prod 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in sinTheta_i around 0 99.6%

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

    1. exp-diff 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 8: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_i around 0 99.6%

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

5. Taylor expanded in sinTheta_O around 0 99.6%

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

6. Taylor expanded in v around 0 97.7%

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

7. Final simplification 97.7%

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

Alternative 9: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

4. Taylor expanded in cosTheta_i around 0 99.6%

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

   1. exp-diff 99.6%

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

   2. frac-2neg 99.6%

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

   3. div-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. div-inv 99.3%

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

   6. distribute-rgt-out 99.6%

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

6. Applied egg-rr 99.6%

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

   1. exp-prod 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in sinTheta_i around 0 99.6%

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

    1. exp-diff 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in v around 0 97.7%

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

13. Final simplification 97.7%

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

Alternative 10: 98.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. *-commutative 99.7%

      \[\leadsto e^{\left(\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   6. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in sinTheta_i around 0 99.6%

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

5. Taylor expanded in cosTheta_O around 0 99.6%

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

6. Taylor expanded in v around 0 97.5%

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

7. Final simplification 97.5%

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

Alternative 11: 6.3% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{cosTheta_O}{\frac{v}{cosTheta_i}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) + (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. Step-by-step derivation

   1. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. *-commutative 99.7%

      \[\leadsto e^{\left(\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   6. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in sinTheta_i around 0 99.6%

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

5. Taylor expanded in cosTheta_O around inf 13.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v}}} \]

6. Taylor expanded in cosTheta_O around 0 6.2%

   \[\leadsto \color{blue}{1 + \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
7. Step-by-step derivation

   1. associate-/l* 6.2%

      \[\leadsto 1 + \color{blue}{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]

8. Simplified 6.2%

   \[\leadsto \color{blue}{1 + \frac{cosTheta_O}{\frac{v}{cosTheta_i}}} \]

9. Final simplification 6.2%

   \[\leadsto 1 + \frac{cosTheta_O}{\frac{v}{cosTheta_i}} \]

Alternative 12: 6.3% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{cosTheta_i \cdot cosTheta_O}{v} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. *-commutative 99.7%

      \[\leadsto e^{\left(\frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate--l- 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   6. associate-+l+ 99.7%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_O \cdot cosTheta_i}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in sinTheta_i around 0 99.6%

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

5. Taylor expanded in cosTheta_O around inf 13.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v}}} \]

6. Taylor expanded in cosTheta_O around 0 6.2%

   \[\leadsto \color{blue}{1 + \frac{cosTheta_O \cdot cosTheta_i}{v}} \]

7. Final simplification 6.2%

   \[\leadsto 1 + \frac{cosTheta_i \cdot cosTheta_O}{v} \]

Reproduce

herbie shell --seed 2023301 
(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))))))