HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 9.1s

Alternatives: 8

Speedup: N/A×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-+l- 99.9%

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

   4. associate-+l- 99.9%

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

   5. sub-neg 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/r* 99.9%

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

   9. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around 0 99.9%

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

5. Taylor expanded in cosTheta_i around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. exp-sum 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. mul-1-neg 99.9%

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

   2. associate-*l/ 99.9%

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

   3. distribute-lft-neg-out 99.9%

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

   4. associate-*l/ 99.9%

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

   5. *-commutative 99.9%

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

   6. +-commutative 99.9%

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

   7. cancel-sign-sub-inv 99.9%

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

   8. *-commutative 99.9%

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

   9. associate-*l/ 99.9%

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

   10. associate-*l/ 99.9%

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

   11. div-sub 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. associate--r+ 99.9%

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

   2. associate-*l/ 99.9%

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

9. Simplified 99.9%

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

10. Taylor expanded in sinTheta_i around 0 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 98.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-+l- 99.9%

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

   4. associate-+l- 99.9%

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

   5. sub-neg 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/r* 99.9%

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

   9. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around 0 99.9%

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

5. Taylor expanded in cosTheta_i around 0 99.9%

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

6. Taylor expanded in v around 0 99.0%

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

7. Final simplification 99.0%

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

Alternative 4: 57.7% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -5.605193857299268 \cdot 10^{-45} \lor \neg \left(sinTheta_i \cdot sinTheta_O \leq 4.203895392974451 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (or (<= (* sinTheta_i sinTheta_O) -5.605193857299268e-45)
         (not (<= (* sinTheta_i sinTheta_O) 4.203895392974451e-45)))
   (/ (* cosTheta_i cosTheta_O) v)
   (/ (* sinTheta_i (- sinTheta_O)) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (((sinTheta_i * sinTheta_O) <= -5.605193857299268e-45f) || !((sinTheta_i * sinTheta_O) <= 4.203895392974451e-45f)) {
		tmp = (cosTheta_i * cosTheta_O) / v;
	} else {
		tmp = (sinTheta_i * -sinTheta_O) / v;
	}
	return tmp;
}

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
    real(4) :: tmp
    if (((sintheta_i * sintheta_o) <= (-5.605193857299268e-45)) .or. (.not. ((sintheta_i * sintheta_o) <= 4.203895392974451e-45))) then
        tmp = (costheta_i * costheta_o) / v
    else
        tmp = (sintheta_i * -sintheta_o) / v
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if ((Float32(sinTheta_i * sinTheta_O) <= Float32(-5.605193857299268e-45)) || !(Float32(sinTheta_i * sinTheta_O) <= Float32(4.203895392974451e-45)))
		tmp = Float32(Float32(cosTheta_i * cosTheta_O) / v);
	else
		tmp = Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (((sinTheta_i * sinTheta_O) <= single(-5.605193857299268e-45)) || ~(((sinTheta_i * sinTheta_O) <= single(4.203895392974451e-45))))
		tmp = (cosTheta_i * cosTheta_O) / v;
	else
		tmp = (sinTheta_i * -sinTheta_O) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -5.60519e-45 or 4.2039e-45 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l- 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub-neg 99.9%

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

      6. associate--l- 99.9%

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

      7. associate-/l* 99.9%

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

      8. associate-/r* 99.9%

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

      9. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in sinTheta_i around 0 99.9%

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

   5. Taylor expanded in cosTheta_i around inf 13.7%

      \[\leadsto e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}} \]
   6. Step-by-step derivation

      1. *-commutative 13.7%

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

      2. associate-/l* 13.7%

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

   7. Simplified 13.7%

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

   8. Taylor expanded in cosTheta_O around 0 6.2%

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

   9. Taylor expanded in cosTheta_i around inf 42.1%

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

   if -5.60519e-45 < (*.f32 sinTheta_i sinTheta_O) < 4.2039e-45

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l- 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. associate-/l* 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in sinTheta_i around inf 6.4%

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

      1. mul-1-neg 6.4%

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

      2. associate-*l/ 6.4%

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

      3. distribute-lft-neg-out 6.4%

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

      4. *-commutative 6.4%

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

      5. distribute-neg-frac 6.4%

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

   6. Simplified 6.4%

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

   7. Taylor expanded in sinTheta_O around 0 6.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
   8. Step-by-step derivation

      1. fma-def 6.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{sinTheta_i \cdot sinTheta_O}{v}, 1\right)} \]

      2. associate-/l* 6.4%

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}, 1\right) \]

      3. fma-def 6.4%

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

      4. neg-mul-1 6.4%

         \[\leadsto \color{blue}{\left(-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)} + 1 \]

      5. +-commutative 6.4%

         \[\leadsto \color{blue}{1 + \left(-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)} \]

      6. unsub-neg 6.4%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}} \]

      7. associate-/l* 6.4%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   9. Simplified 6.4%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   10. Taylor expanded in sinTheta_i around inf 95.7%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]
   11. Step-by-step derivation

       1. associate-*r/ 95.7%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}} \]

       2. mul-1-neg 95.7%

          \[\leadsto \frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} \]

       3. distribute-rgt-neg-in 95.7%

          \[\leadsto \frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v} \]

   12. Simplified 95.7%

       \[\leadsto \color{blue}{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -5.605193857299268 \cdot 10^{-45} \lor \neg \left(sinTheta_i \cdot sinTheta_O \leq 4.203895392974451 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}\\ \end{array} \]

Alternative 5: 45.6% accurate, 16.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.401298464324817 \cdot 10^{-45} \lor \neg \left(cosTheta_i \cdot cosTheta_O \leq 0\right):\\ \;\;\;\;sinTheta_O \cdot \frac{sinTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (or (<= (* cosTheta_i cosTheta_O) -1.401298464324817e-45)
         (not (<= (* cosTheta_i cosTheta_O) 0.0)))
   (* sinTheta_O (/ sinTheta_i v))
   (/ (* cosTheta_i cosTheta_O) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (((cosTheta_i * cosTheta_O) <= -1.401298464324817e-45f) || !((cosTheta_i * cosTheta_O) <= 0.0f)) {
		tmp = sinTheta_O * (sinTheta_i / v);
	} else {
		tmp = (cosTheta_i * cosTheta_O) / v;
	}
	return tmp;
}

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
    real(4) :: tmp
    if (((costheta_i * costheta_o) <= (-1.401298464324817e-45)) .or. (.not. ((costheta_i * costheta_o) <= 0.0e0))) then
        tmp = sintheta_o * (sintheta_i / v)
    else
        tmp = (costheta_i * costheta_o) / v
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if ((Float32(cosTheta_i * cosTheta_O) <= Float32(-1.401298464324817e-45)) || !(Float32(cosTheta_i * cosTheta_O) <= Float32(0.0)))
		tmp = Float32(sinTheta_O * Float32(sinTheta_i / v));
	else
		tmp = Float32(Float32(cosTheta_i * cosTheta_O) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (((cosTheta_i * cosTheta_O) <= single(-1.401298464324817e-45)) || ~(((cosTheta_i * cosTheta_O) <= single(0.0))))
		tmp = sinTheta_O * (sinTheta_i / v);
	else
		tmp = (cosTheta_i * cosTheta_O) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -1.4013e-45 or 0.0 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l- 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. associate-/l* 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in sinTheta_i around inf 19.0%

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

      1. mul-1-neg 19.0%

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

      2. associate-*l/ 19.0%

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

      3. distribute-lft-neg-out 19.0%

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

      4. *-commutative 19.0%

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

      5. distribute-neg-frac 19.0%

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

   6. Simplified 19.0%

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

   7. Taylor expanded in sinTheta_O around 0 6.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
   8. Step-by-step derivation

      1. fma-def 6.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{sinTheta_i \cdot sinTheta_O}{v}, 1\right)} \]

      2. associate-/l* 6.1%

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}, 1\right) \]

      3. fma-def 6.1%

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

      4. neg-mul-1 6.1%

         \[\leadsto \color{blue}{\left(-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)} + 1 \]

      5. +-commutative 6.1%

         \[\leadsto \color{blue}{1 + \left(-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)} \]

      6. unsub-neg 6.1%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}} \]

      7. associate-/l* 6.1%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   9. Simplified 6.1%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   10. Taylor expanded in sinTheta_i around inf 36.7%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]
   11. Step-by-step derivation

       1. associate-*r/ 36.7%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}} \]

       2. mul-1-neg 36.7%

          \[\leadsto \frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} \]

       3. distribute-rgt-neg-in 36.7%

          \[\leadsto \frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v} \]

   12. Simplified 36.7%

       \[\leadsto \color{blue}{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}} \]
   13. Step-by-step derivation

       1. *-commutative 36.7%

          \[\leadsto \frac{\color{blue}{\left(-sinTheta_O\right) \cdot sinTheta_i}}{v} \]

       2. associate-*r/ 23.3%

          \[\leadsto \color{blue}{\left(-sinTheta_O\right) \cdot \frac{sinTheta_i}{v}} \]

       3. *-commutative 23.3%

          \[\leadsto \color{blue}{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)} \]

       4. add-sqr-sqrt 10.3%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \color{blue}{\left(\sqrt{-sinTheta_O} \cdot \sqrt{-sinTheta_O}\right)} \]

       5. sqrt-unprod 42.1%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \color{blue}{\sqrt{\left(-sinTheta_O\right) \cdot \left(-sinTheta_O\right)}} \]

       6. sqr-neg 42.1%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \sqrt{\color{blue}{sinTheta_O \cdot sinTheta_O}} \]

       7. sqrt-unprod 13.0%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \color{blue}{\left(\sqrt{sinTheta_O} \cdot \sqrt{sinTheta_O}\right)} \]

       8. add-sqr-sqrt 23.3%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \color{blue}{sinTheta_O} \]

   14. Applied egg-rr 23.3%

       \[\leadsto \color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O} \]

   if -1.4013e-45 < (*.f32 cosTheta_i cosTheta_O) < 0.0

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l- 99.8%

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

      4. associate-+l- 99.8%

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

      5. sub-neg 99.8%

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

      6. associate--l- 99.8%

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

      7. associate-/l* 99.8%

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

      8. associate-/r* 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in sinTheta_i around 0 99.8%

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

   5. Taylor expanded in cosTheta_i around inf 6.5%

      \[\leadsto e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}} \]
   6. Step-by-step derivation

      1. *-commutative 6.5%

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

      2. associate-/l* 6.5%

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

   7. Simplified 6.5%

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

   8. Taylor expanded in cosTheta_O around 0 6.5%

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

   9. Taylor expanded in cosTheta_i around inf 97.2%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.401298464324817 \cdot 10^{-45} \lor \neg \left(cosTheta_i \cdot cosTheta_O \leq 0\right):\\ \;\;\;\;sinTheta_O \cdot \frac{sinTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \]

Alternative 6: 22.2% accurate, 31.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta_i \leq -3.0000001167615996 \cdot 10^{-16}:\\ \;\;\;\;cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;sinTheta_O \cdot \frac{sinTheta_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_i -3.0000001167615996e-16)
   (* cosTheta_O (/ cosTheta_i v))
   (* sinTheta_O (/ sinTheta_i v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_i <= -3.0000001167615996e-16f) {
		tmp = cosTheta_O * (cosTheta_i / v);
	} else {
		tmp = sinTheta_O * (sinTheta_i / v);
	}
	return tmp;
}

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
    real(4) :: tmp
    if (sintheta_i <= (-3.0000001167615996e-16)) then
        tmp = costheta_o * (costheta_i / v)
    else
        tmp = sintheta_o * (sintheta_i / v)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(-3.0000001167615996e-16))
		tmp = Float32(cosTheta_O * Float32(cosTheta_i / v));
	else
		tmp = Float32(sinTheta_O * Float32(sinTheta_i / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_i <= single(-3.0000001167615996e-16))
		tmp = cosTheta_O * (cosTheta_i / v);
	else
		tmp = sinTheta_O * (sinTheta_i / v);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < -3.0000001e-16

   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. 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. sub-neg 99.6%

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

      3. associate-+l- 99.6%

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

      4. associate-+l- 99.6%

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

      5. sub-neg 99.6%

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

      6. associate--l- 99.6%

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

      7. associate-/l* 99.6%

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

      8. associate-/r* 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in sinTheta_i around 0 99.6%

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

   5. Taylor expanded in cosTheta_i around inf 17.2%

      \[\leadsto e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}} \]
   6. Step-by-step derivation

      1. *-commutative 17.2%

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

      2. associate-/l* 17.2%

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

   7. Simplified 17.2%

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

   8. Taylor expanded in cosTheta_O around 0 6.3%

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

   9. Taylor expanded in cosTheta_i around inf 37.4%

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

       1. associate-*l/ 15.8%

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

   11. Simplified 15.8%

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

   if -3.0000001e-16 < sinTheta_i

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l- 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--l- 100.0%

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

      7. associate-/l* 100.0%

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

      8. associate-/r* 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in sinTheta_i around inf 15.5%

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

      1. mul-1-neg 15.5%

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

      2. associate-*l/ 15.5%

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

      3. distribute-lft-neg-out 15.5%

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

      4. *-commutative 15.5%

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

      5. distribute-neg-frac 15.5%

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

   6. Simplified 15.5%

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

   7. Taylor expanded in sinTheta_O around 0 6.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + 1} \]
   8. Step-by-step derivation

      1. fma-def 6.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{sinTheta_i \cdot sinTheta_O}{v}, 1\right)} \]

      2. associate-/l* 6.2%

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}, 1\right) \]

      3. fma-def 6.2%

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

      4. neg-mul-1 6.2%

         \[\leadsto \color{blue}{\left(-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)} + 1 \]

      5. +-commutative 6.2%

         \[\leadsto \color{blue}{1 + \left(-\frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)} \]

      6. unsub-neg 6.2%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_i}{\frac{v}{sinTheta_O}}} \]

      7. associate-/l* 6.2%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   9. Simplified 6.2%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_i \cdot sinTheta_O}{v}} \]

   10. Taylor expanded in sinTheta_i around inf 41.0%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}} \]
   11. Step-by-step derivation

       1. associate-*r/ 41.0%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}} \]

       2. mul-1-neg 41.0%

          \[\leadsto \frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} \]

       3. distribute-rgt-neg-in 41.0%

          \[\leadsto \frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v} \]

   12. Simplified 41.0%

       \[\leadsto \color{blue}{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}} \]
   13. Step-by-step derivation

       1. *-commutative 41.0%

          \[\leadsto \frac{\color{blue}{\left(-sinTheta_O\right) \cdot sinTheta_i}}{v} \]

       2. associate-*r/ 24.3%

          \[\leadsto \color{blue}{\left(-sinTheta_O\right) \cdot \frac{sinTheta_i}{v}} \]

       3. *-commutative 24.3%

          \[\leadsto \color{blue}{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)} \]

       4. add-sqr-sqrt 10.3%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \color{blue}{\left(\sqrt{-sinTheta_O} \cdot \sqrt{-sinTheta_O}\right)} \]

       5. sqrt-unprod 46.6%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \color{blue}{\sqrt{\left(-sinTheta_O\right) \cdot \left(-sinTheta_O\right)}} \]

       6. sqr-neg 46.6%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \sqrt{\color{blue}{sinTheta_O \cdot sinTheta_O}} \]

       7. sqrt-unprod 14.0%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \color{blue}{\left(\sqrt{sinTheta_O} \cdot \sqrt{sinTheta_O}\right)} \]

       8. add-sqr-sqrt 24.3%

          \[\leadsto \frac{sinTheta_i}{v} \cdot \color{blue}{sinTheta_O} \]

   14. Applied egg-rr 24.3%

       \[\leadsto \color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_i \leq -3.0000001167615996 \cdot 10^{-16}:\\ \;\;\;\;cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \mathbf{else}:\\ \;\;\;\;sinTheta_O \cdot \frac{sinTheta_i}{v}\\ \end{array} \]

Alternative 7: 20.3% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-+l- 99.9%

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

   4. associate-+l- 99.9%

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

   5. sub-neg 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/r* 99.9%

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

   9. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around 0 99.9%

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

5. Taylor expanded in cosTheta_i around inf 13.0%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}} \]
6. Step-by-step derivation

   1. *-commutative 13.0%

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

   2. associate-/l* 13.0%

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

7. Simplified 13.0%

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

8. Taylor expanded in cosTheta_O around 0 6.2%

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

9. Taylor expanded in cosTheta_i around inf 42.1%

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

    1. associate-*l/ 19.5%

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

11. Simplified 19.5%

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

12. Final simplification 19.5%

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

Alternative 8: 6.4% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

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
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-+l- 99.9%

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

   4. associate-+l- 99.9%

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

   5. sub-neg 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/r* 99.9%

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

   9. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in sinTheta_i around inf 19.1%

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

   1. mul-1-neg 19.1%

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

   2. associate-*l/ 19.1%

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

   3. distribute-lft-neg-out 19.1%

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

   4. *-commutative 19.1%

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

   5. distribute-neg-frac 19.1%

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

6. Simplified 19.1%

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

7. Taylor expanded in sinTheta_O around 0 6.4%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023227 
(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))))))