HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.6%

Time: 18.7s

Alternatives: 9

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 99.6%

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

   2. fma-define 99.6%

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

   3. prod-exp 99.7%

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

   4. add-exp-log 99.7%

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

   5. *-commutative 99.7%

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

   6. exp-sum 99.7%

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

   7. fma-define 99.7%

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

   8. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in cosTheta_i around inf 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 99.6%

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

   2. fma-define 99.6%

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

   3. prod-exp 99.7%

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

   4. add-exp-log 99.7%

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

   5. *-commutative 99.7%

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

   6. exp-sum 99.7%

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

   7. fma-define 99.7%

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

   8. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in cosTheta_i around inf 99.7%

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

8. Taylor expanded in cosTheta_O around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. distribute-neg-frac 99.7%

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

   3. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. exp-sum 99.7%

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

   2. *-commutative 99.7%

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

   3. rem-exp-log 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. associate--l- 99.7%

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

   7. associate-/l* 99.7%

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

   8. associate-/l* 99.7%

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

   9. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 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 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 99.6%

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

   2. fma-define 99.6%

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

   3. prod-exp 99.7%

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

   4. add-exp-log 99.7%

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

   5. *-commutative 99.7%

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

   6. exp-sum 99.7%

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

   7. fma-define 99.7%

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

   8. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in cosTheta_i around inf 99.7%

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

8. Taylor expanded in cosTheta_O around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. distribute-neg-frac 99.7%

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

   3. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in v around 0 99.7%

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

    1. metadata-eval 99.7%

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

    2. distribute-neg-frac 99.7%

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

    3. *-commutative 99.7%

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

    4. prod-exp 99.7%

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

    5. distribute-neg-frac 99.7%

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

    6. metadata-eval 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 5: 97.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around 0 99.6%

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

   1. associate-+r+ 99.6%

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

   2. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in v around 0 97.5%

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

9. Final simplification 97.5%

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

Alternative 6: 70.7% accurate, 24.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return -1.0f + (1.0f - (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 = (-1.0e0) + (1.0e0 - (sintheta_o * (sintheta_i / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 10.6%

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

   1. *-commutative 10.6%

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

   2. associate-*r/ 10.6%

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

   3. neg-mul-1 10.6%

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

   4. distribute-lft-neg-in 10.6%

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

7. Simplified 10.6%

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

8. Taylor expanded in sinTheta_i around 0 6.2%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]
9. Step-by-step derivation

   1. mul-1-neg 6.2%

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

   2. unsub-neg 6.2%

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

   3. associate-/l* 6.2%

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

10. Simplified 6.2%

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

11. Taylor expanded in sinTheta_O around inf 43.0%

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

    1. mul-1-neg 43.0%

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

    2. associate-*r/ 22.6%

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

    3. distribute-lft-neg-out 22.6%

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

    4. *-commutative 22.6%

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

13. Simplified 22.6%

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

    1. expm1-log1p-u 22.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta\_i}{v} \cdot \left(-sinTheta\_O\right)\right)\right)} \]

    2. expm1-undefine 76.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{sinTheta\_i}{v} \cdot \left(-sinTheta\_O\right)\right)} - 1} \]

15. Applied egg-rr 76.7%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{sinTheta\_i}{v} \cdot \left(-sinTheta\_O\right)\right)} - 1} \]
16. Step-by-step derivation

    1. sub-neg 76.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{sinTheta\_i}{v} \cdot \left(-sinTheta\_O\right)\right)} + \left(-1\right)} \]

    2. log1p-undefine 76.7%

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

    3. distribute-rgt-neg-in 76.7%

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

    4. neg-mul-1 76.7%

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

    5. *-commutative 76.7%

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

    6. associate-/l* 76.7%

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

    7. rem-exp-log 77.0%

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

    8. mul-1-neg 77.0%

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

    9. unsub-neg 77.0%

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

    10. associate-/l* 77.0%

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

    11. metadata-eval 77.0%

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

17. Simplified 77.0%

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

18. Final simplification 77.0%

    \[\leadsto -1 + \left(1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right) \]
19. Add Preprocessing

Alternative 7: 19.8% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 10.6%

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

   1. *-commutative 10.6%

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

   2. associate-*r/ 10.6%

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

   3. neg-mul-1 10.6%

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

   4. distribute-lft-neg-in 10.6%

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

7. Simplified 10.6%

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

8. Taylor expanded in sinTheta_i around 0 6.2%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]
9. Step-by-step derivation

   1. mul-1-neg 6.2%

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

   2. unsub-neg 6.2%

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

   3. associate-/l* 6.2%

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

10. Simplified 6.2%

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

11. Taylor expanded in sinTheta_O around inf 43.0%

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

    1. mul-1-neg 43.0%

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

    2. associate-*r/ 22.6%

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

    3. distribute-lft-neg-out 22.6%

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

    4. *-commutative 22.6%

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

13. Simplified 22.6%

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

14. Taylor expanded in sinTheta_i around 0 43.0%

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

    1. mul-1-neg 43.0%

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

    2. distribute-frac-neg 43.0%

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

    3. *-commutative 43.0%

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

    4. distribute-rgt-neg-out 43.0%

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

    5. associate-/l* 22.6%

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

16. Simplified 22.6%

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

17. Final simplification 22.6%

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

Alternative 8: 19.8% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 10.6%

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

   1. *-commutative 10.6%

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

   2. associate-*r/ 10.6%

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

   3. neg-mul-1 10.6%

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

   4. distribute-lft-neg-in 10.6%

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

7. Simplified 10.6%

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

8. Taylor expanded in sinTheta_i around 0 6.2%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]
9. Step-by-step derivation

   1. mul-1-neg 6.2%

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

   2. unsub-neg 6.2%

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

   3. associate-/l* 6.2%

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

10. Simplified 6.2%

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

11. Taylor expanded in sinTheta_O around inf 43.0%

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

    1. mul-1-neg 43.0%

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

    2. associate-*r/ 22.6%

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

    3. distribute-lft-neg-out 22.6%

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

    4. *-commutative 22.6%

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

13. Simplified 22.6%

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

14. Final simplification 22.6%

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

Alternative 9: 38.3% accurate, 37.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around inf 10.6%

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

   1. *-commutative 10.6%

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

   2. associate-*r/ 10.6%

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

   3. neg-mul-1 10.6%

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

   4. distribute-lft-neg-in 10.6%

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

7. Simplified 10.6%

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

8. Taylor expanded in sinTheta_i around 0 6.2%

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}} \]
9. Step-by-step derivation

   1. mul-1-neg 6.2%

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

   2. unsub-neg 6.2%

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

   3. associate-/l* 6.2%

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

10. Simplified 6.2%

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

11. Taylor expanded in sinTheta_O around inf 43.0%

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

    1. mul-1-neg 43.0%

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

    2. associate-*r/ 22.6%

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

    3. distribute-lft-neg-out 22.6%

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

    4. *-commutative 22.6%

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

13. Simplified 22.6%

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

    1. associate-*l/ 43.0%

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

15. Applied egg-rr 43.0%

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

16. Final simplification 43.0%

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

Reproduce

herbie shell --seed 2024044 
(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))))))