HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 12.5s

Alternatives: 8

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, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (exp (/ (+ (* 0.6931 v) -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 * v) + -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 * v) + (-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(Float32(Float32(0.6931) * v) + 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) * v) + single(-1.0)) / v)) / v);
end

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. associate-/l* 99.5%

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

   8. associate-/l* 99.5%

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

   9. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Taylor expanded in v around 0 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 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 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. associate-/l* 99.5%

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

   8. associate-/l* 99.5%

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

   9. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 97.9% 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 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. associate-/l* 99.5%

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

   8. associate-/l* 99.5%

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

   9. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Taylor expanded in v around 0 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 11.4% accurate, 24.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * sinTheta_i) * ((sinTheta_O / 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 * sintheta_i) * ((sintheta_o / v) / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. associate-/l* 99.5%

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

   8. associate-/l* 99.5%

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

   9. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in sinTheta_i around inf 14.2%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. associate-*r/ 14.2%

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

   2. mul-1-neg 14.2%

      \[\leadsto \frac{0.5}{v} \cdot e^{\frac{\color{blue}{-sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   3. distribute-lft-neg-out 14.2%

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

   4. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in v around inf 4.4%

   \[\leadsto \color{blue}{\frac{0.5 + -0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}} \]

9. Taylor expanded in sinTheta_O around inf 41.2%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{v} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 39.3%

       \[\leadsto \frac{\color{blue}{\sqrt{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \cdot \sqrt{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{v} \]

    2. sqrt-unprod 52.7%

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

    3. swap-sqr 52.7%

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

    4. metadata-eval 52.7%

       \[\leadsto \frac{\sqrt{\color{blue}{0.25} \cdot \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v} \]

    5. metadata-eval 52.7%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(0.5 \cdot 0.5\right)} \cdot \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v} \]

    6. swap-sqr 52.7%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) \cdot \left(0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}}{v} \]

    7. *-commutative 52.7%

       \[\leadsto \frac{\sqrt{\left(0.5 \cdot \frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}\right) \cdot \left(0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v} \]

    8. associate-*r/ 52.7%

       \[\leadsto \frac{\sqrt{\left(0.5 \cdot \color{blue}{\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}\right) \cdot \left(0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v} \]

    9. *-commutative 52.7%

       \[\leadsto \frac{\sqrt{\left(0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \cdot \left(0.5 \cdot \frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}\right)}}{v} \]

    10. associate-*r/ 48.3%

        \[\leadsto \frac{\sqrt{\left(0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \cdot \left(0.5 \cdot \color{blue}{\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}\right)}}{v} \]

    11. sqrt-unprod 13.0%

        \[\leadsto \frac{\color{blue}{\sqrt{0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)} \cdot \sqrt{0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}}}{v} \]

    12. add-sqr-sqrt 16.0%

        \[\leadsto \frac{\color{blue}{0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}}{v} \]

    13. associate-*r* 15.3%

        \[\leadsto \frac{\color{blue}{\left(0.5 \cdot sinTheta\_i\right) \cdot \frac{sinTheta\_O}{v}}}{v} \]

    14. associate-/l* 13.4%

        \[\leadsto \color{blue}{\left(0.5 \cdot sinTheta\_i\right) \cdot \frac{\frac{sinTheta\_O}{v}}{v}} \]

    15. *-commutative 13.4%

        \[\leadsto \color{blue}{\left(sinTheta\_i \cdot 0.5\right)} \cdot \frac{\frac{sinTheta\_O}{v}}{v} \]

11. Applied egg-rr 13.4%

    \[\leadsto \color{blue}{\left(sinTheta\_i \cdot 0.5\right) \cdot \frac{\frac{sinTheta\_O}{v}}{v}} \]

12. Final simplification 13.4%

    \[\leadsto \left(0.5 \cdot sinTheta\_i\right) \cdot \frac{\frac{sinTheta\_O}{v}}{v} \]
13. Add Preprocessing

Alternative 5: 14.1% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. associate-/l* 99.5%

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

   8. associate-/l* 99.5%

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

   9. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in sinTheta_i around inf 14.2%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. associate-*r/ 14.2%

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

   2. mul-1-neg 14.2%

      \[\leadsto \frac{0.5}{v} \cdot e^{\frac{\color{blue}{-sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   3. distribute-lft-neg-out 14.2%

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

   4. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in v around inf 4.4%

   \[\leadsto \color{blue}{\frac{0.5 + -0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}} \]

9. Taylor expanded in sinTheta_O around inf 41.2%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{v} \]
10. Step-by-step derivation

    1. *-commutative 41.2%

       \[\leadsto \frac{\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v} \cdot -0.5}}{v} \]

    2. associate-/l* 40.8%

       \[\leadsto \color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v} \cdot \frac{-0.5}{v}} \]

    3. *-commutative 40.8%

       \[\leadsto \frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v} \cdot \frac{-0.5}{v} \]

    4. associate-*r/ 15.4%

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

11. Applied egg-rr 15.4%

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

12. Final simplification 15.4%

    \[\leadsto \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) \cdot \frac{-0.5}{v} \]
13. Add Preprocessing

Alternative 6: 36.5% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (-0.5f * ((sinTheta_O * sinTheta_i) / 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) * ((sintheta_o * sintheta_i) / v)) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. associate-/l* 99.5%

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

   8. associate-/l* 99.5%

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

   9. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in sinTheta_i around inf 14.2%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. associate-*r/ 14.2%

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

   2. mul-1-neg 14.2%

      \[\leadsto \frac{0.5}{v} \cdot e^{\frac{\color{blue}{-sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   3. distribute-lft-neg-out 14.2%

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

   4. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in v around inf 4.4%

   \[\leadsto \color{blue}{\frac{0.5 + -0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}} \]

9. Taylor expanded in sinTheta_O around inf 41.2%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{v} \]

10. Final simplification 41.2%

    \[\leadsto \frac{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v} \]
11. Add Preprocessing

Alternative 7: 37.0% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}}{v} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. associate-/l* 99.5%

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

   8. associate-/l* 99.5%

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

   9. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in sinTheta_i around inf 14.2%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. associate-*r/ 14.2%

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

   2. mul-1-neg 14.2%

      \[\leadsto \frac{0.5}{v} \cdot e^{\frac{\color{blue}{-sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   3. distribute-lft-neg-out 14.2%

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

   4. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in v around inf 4.4%

   \[\leadsto \color{blue}{\frac{0.5 + -0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}} \]

9. Taylor expanded in sinTheta_O around inf 41.2%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{v} \]
10. Step-by-step derivation

    1. associate-*r/ 16.1%

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

    2. associate-*l* 15.4%

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

    3. rem-square-sqrt 12.1%

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

    4. fabs-sqr 12.1%

       \[\leadsto \frac{\color{blue}{\left|\sqrt{\left(-0.5 \cdot sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}} \cdot \sqrt{\left(-0.5 \cdot sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}\right|}}{v} \]

    5. rem-square-sqrt 15.4%

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

    6. associate-*l* 16.1%

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

    7. fabs-mul 16.1%

       \[\leadsto \frac{\color{blue}{\left|-0.5\right| \cdot \left|sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right|}}{v} \]

    8. metadata-eval 16.1%

       \[\leadsto \frac{\color{blue}{0.5} \cdot \left|sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right|}{v} \]

    9. rem-square-sqrt 12.6%

       \[\leadsto \frac{0.5 \cdot \left|\color{blue}{\sqrt{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \cdot \sqrt{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}}\right|}{v} \]

    10. fabs-sqr 12.6%

        \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\sqrt{sinTheta\_O \cdot \frac{sinTheta\_i}{v}} \cdot \sqrt{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\right)}}{v} \]

    11. rem-square-sqrt 16.0%

        \[\leadsto \frac{0.5 \cdot \color{blue}{\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)}}{v} \]

    12. *-commutative 16.0%

        \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\frac{sinTheta\_i}{v} \cdot sinTheta\_O\right)}}{v} \]

    13. associate-*l/ 41.2%

        \[\leadsto \frac{0.5 \cdot \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{v} \]

    14. associate-*r/ 16.0%

        \[\leadsto \frac{0.5 \cdot \color{blue}{\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}}{v} \]

11. Simplified 16.0%

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

    1. add-sqr-sqrt 13.0%

       \[\leadsto \frac{\color{blue}{\sqrt{0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)} \cdot \sqrt{0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}}}{v} \]

    2. sqrt-unprod 48.3%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \cdot \left(0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right)}}}{v} \]

    3. associate-*r/ 52.7%

       \[\leadsto \frac{\sqrt{\left(0.5 \cdot \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot \left(0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right)}}{v} \]

    4. *-commutative 52.7%

       \[\leadsto \frac{\sqrt{\left(0.5 \cdot \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}\right) \cdot \left(0.5 \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right)}}{v} \]

    5. associate-*r/ 52.7%

       \[\leadsto \frac{\sqrt{\left(0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) \cdot \left(0.5 \cdot \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)}}{v} \]

    6. *-commutative 52.7%

       \[\leadsto \frac{\sqrt{\left(0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) \cdot \left(0.5 \cdot \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v}\right)}}{v} \]

    7. swap-sqr 52.7%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}}{v} \]

    8. metadata-eval 52.7%

       \[\leadsto \frac{\sqrt{\color{blue}{0.25} \cdot \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}}{v} \]

    9. metadata-eval 52.7%

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

    10. swap-sqr 52.7%

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

    11. sqrt-unprod 39.3%

        \[\leadsto \frac{\color{blue}{\sqrt{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \cdot \sqrt{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{v} \]

    12. add-sqr-sqrt 41.2%

        \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{v} \]

    13. clear-num 41.8%

        \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{1}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}}}{v} \]

    14. un-div-inv 41.8%

        \[\leadsto \frac{\color{blue}{\frac{-0.5}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}}}{v} \]

13. Applied egg-rr 41.8%

    \[\leadsto \frac{\color{blue}{\frac{-0.5}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}}}{v} \]

14. Final simplification 41.8%

    \[\leadsto \frac{\frac{-0.5}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}}{v} \]
15. Add Preprocessing

Alternative 8: 4.7% accurate, 74.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 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 = 0.5e0 / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. associate-/l* 99.5%

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

   8. associate-/l* 99.5%

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

   9. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in sinTheta_i around inf 14.2%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. associate-*r/ 14.2%

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

   2. mul-1-neg 14.2%

      \[\leadsto \frac{0.5}{v} \cdot e^{\frac{\color{blue}{-sinTheta\_O \cdot sinTheta\_i}}{v}} \]

   3. distribute-lft-neg-out 14.2%

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

   4. *-commutative 14.2%

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

7. Simplified 14.2%

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

8. Taylor expanded in v around inf 4.5%

   \[\leadsto \color{blue}{\frac{0.5}{v}} \]

9. Final simplification 4.5%

   \[\leadsto \frac{0.5}{v} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024051 
(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))))))