HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 11.5s

Alternatives: 12

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (exp(single(0.6931)) * exp(((single(-1.0) / v) - log(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. Step-by-step derivation

   1. add-exp-log 100.0%

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

   2. log-div 100.0%

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

   3. add-log-exp 100.0%

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

   4. sub-neg 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. associate--l+ 100.0%

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

   2. exp-sum 100.0%

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

10. Applied egg-rr 100.0%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (exp (- (- 0.6931 (/ 1.0 v)) (log 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)) - logf(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)) - log(v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * exp(Float32(Float32(Float32(0.6931) - Float32(Float32(1.0) / v)) - log(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)) - log(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. Step-by-step derivation

   1. add-exp-log 100.0%

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

   2. log-div 100.0%

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

   3. add-log-exp 100.0%

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

   4. sub-neg 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto 0.5 \cdot e^{\left(0.6931 - \frac{1}{v}\right) - \log v} \]
10. Add Preprocessing

Alternative 3: 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. Add Preprocessing

Alternative 4: 37.6% accurate, 24.8× speedup

Math

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

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(Float32(0.5) * Float32(sinTheta_i * 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 12.7%

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

   1. associate-*r/ 12.7%

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

   2. mul-1-neg 12.7%

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

7. Simplified 12.7%

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

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 38.7%

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

    1. associate-*r/ 13.6%

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

    2. *-commutative 13.6%

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

    3. associate-*l/ 38.7%

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

    4. associate-*r/ 13.6%

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

11. Simplified 13.6%

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

    1. associate-*r/ 38.7%

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

    2. *-commutative 38.7%

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

    3. add-sqr-sqrt 36.8%

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

    4. sqrt-unprod 49.8%

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

    5. sqr-neg 49.8%

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

    6. mul-1-neg 49.8%

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

    7. mul-1-neg 49.8%

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

    8. sqrt-unprod 36.8%

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

    9. add-sqr-sqrt 38.7%

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

    10. mul-1-neg 38.7%

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

    11. distribute-rgt-neg-in 38.7%

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

    12. associate-*r/ 40.0%

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

    13. associate-*r* 39.0%

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

    14. *-commutative 39.0%

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

    15. distribute-neg-frac 39.0%

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

    16. *-commutative 39.0%

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

    17. associate-*r* 40.0%

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

    18. distribute-lft-neg-in 40.0%

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

    19. metadata-eval 40.0%

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

    20. *-commutative 40.0%

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

13. Applied egg-rr 40.0%

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

Alternative 5: 37.2% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(-0.5) / (v / (sinTheta_i * sinTheta_O))) / 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 12.7%

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

   1. associate-*r/ 12.7%

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

   2. mul-1-neg 12.7%

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

7. Simplified 12.7%

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

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 38.7%

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

    1. associate-*r/ 13.6%

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

    2. *-commutative 13.6%

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

    3. associate-*l/ 38.7%

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

    4. associate-*r/ 13.6%

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

11. Simplified 13.6%

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

    1. associate-*r/ 38.7%

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

    2. *-commutative 38.7%

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

    3. clear-num 39.3%

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

    4. un-div-inv 39.3%

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

    5. *-commutative 39.3%

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

13. Applied egg-rr 39.3%

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

Alternative 6: 37.2% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(-0.5) / v) / (v / (sinTheta_i * sinTheta_O));
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 12.7%

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

   1. associate-*r/ 12.7%

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

   2. mul-1-neg 12.7%

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

7. Simplified 12.7%

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

8. Taylor expanded in sinTheta_O around 0 4.4%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 4.4%

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

   2. unsub-neg 4.4%

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

   3. associate-*r/ 4.4%

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

10. Simplified 4.4%

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

11. Taylor expanded in sinTheta_O around inf 38.3%

    \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)} \]
12. Step-by-step derivation

    1. associate-*r* 38.3%

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

    2. clear-num 38.9%

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

    3. un-div-inv 38.9%

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

    4. associate-*l/ 38.9%

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

    5. metadata-eval 38.9%

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

    6. *-commutative 38.9%

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

13. Applied egg-rr 38.9%

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

Alternative 7: 36.8% accurate, 24.8× speedup

Math

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

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) * Float32(Float32(sinTheta_i * 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 12.7%

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

   1. associate-*r/ 12.7%

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

   2. mul-1-neg 12.7%

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

7. Simplified 12.7%

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

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 38.7%

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

10. Final simplification 38.7%

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

Alternative 8: 14.2% accurate, 24.8× speedup

Math

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

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) * Float32(sinTheta_i * 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 12.7%

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

   1. associate-*r/ 12.7%

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

   2. mul-1-neg 12.7%

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

7. Simplified 12.7%

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

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 38.7%

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

    1. associate-*r/ 13.6%

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

    2. *-commutative 13.6%

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

    3. associate-*l/ 38.7%

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

    4. associate-*r/ 13.6%

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

11. Simplified 13.6%

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

Alternative 9: 14.0% 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 12.7%

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

   1. associate-*r/ 12.7%

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

   2. mul-1-neg 12.7%

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

7. Simplified 12.7%

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

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 38.7%

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

    1. associate-*r/ 13.6%

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

    2. *-commutative 13.6%

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

    3. associate-*l/ 38.7%

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

    4. associate-*r/ 13.6%

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

11. Simplified 13.6%

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

    1. metadata-eval 13.6%

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

    2. associate-*r/ 38.7%

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

    3. *-commutative 38.7%

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

    4. associate-*r* 38.7%

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

    5. associate-*l/ 38.3%

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

    6. add-sqr-sqrt 36.4%

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

    7. sqrt-unprod 49.4%

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

    8. mul-1-neg 49.4%

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

    9. mul-1-neg 49.4%

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

    10. sqr-neg 49.4%

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

    11. sqrt-unprod 36.4%

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

    12. add-sqr-sqrt 38.3%

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

    13. *-commutative 38.3%

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

    14. associate-*r/ 13.3%

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

13. Applied egg-rr 13.3%

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

14. Final simplification 13.3%

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

Alternative 10: 11.5% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sinTheta_i * ((single(-0.5) / v) * (sinTheta_O / 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 12.7%

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

   1. associate-*r/ 12.7%

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

   2. mul-1-neg 12.7%

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

7. Simplified 12.7%

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

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 38.7%

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

    1. associate-*r/ 13.6%

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

    2. *-commutative 13.6%

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

    3. associate-*l/ 38.7%

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

    4. associate-*r/ 13.6%

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

11. Simplified 13.6%

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

    1. metadata-eval 13.6%

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

    2. associate-*r/ 38.7%

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

    3. *-commutative 38.7%

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

    4. associate-*r* 38.7%

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

    5. associate-*l/ 38.3%

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

    6. associate-*r* 38.3%

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

    7. *-commutative 38.3%

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

    8. associate-*r/ 13.3%

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

    9. *-commutative 13.3%

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

    10. associate-*r* 9.7%

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

    11. associate-*l/ 9.7%

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

    12. metadata-eval 9.7%

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

13. Applied egg-rr 9.7%

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

14. Final simplification 9.7%

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

Alternative 11: 11.5% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sinTheta_i * ((sinTheta_O * (single(-0.5) / 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 12.7%

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

   1. associate-*r/ 12.7%

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

   2. mul-1-neg 12.7%

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

7. Simplified 12.7%

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

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 38.7%

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

    1. associate-*r/ 13.6%

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

    2. *-commutative 13.6%

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

    3. associate-*l/ 38.7%

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

    4. associate-*r/ 13.6%

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

11. Simplified 13.6%

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

    1. associate-*r* 13.6%

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

    2. *-commutative 13.6%

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

    3. associate-/l* 39.0%

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

    4. associate-*r* 39.0%

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

    5. associate-/l* 13.6%

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

    6. associate-/l* 9.7%

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

    7. *-commutative 9.7%

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

    8. associate-/l* 9.7%

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

13. Applied egg-rr 9.7%

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

Alternative 12: 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 12.7%

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

   1. associate-*r/ 12.7%

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

   2. mul-1-neg 12.7%

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

7. Simplified 12.7%

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

8. Taylor expanded in v around inf 4.5%

   \[\leadsto \color{blue}{\frac{0.5}{v}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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))))))