HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 13.2s

Alternatives: 9

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0.6931 + \left(\frac{-1}{v} - \frac{sinTheta_O \cdot sinTheta_i}{v}\right)}\\ \frac{0.5}{v} \cdot \left(t_0 + \frac{cosTheta_O \cdot \left(t_0 \cdot cosTheta_i\right)}{v}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (exp (+ 0.6931 (- (/ -1.0 v) (/ (* sinTheta_O sinTheta_i) v))))))
   (* (/ 0.5 v) (+ t_0 (/ (* cosTheta_O (* t_0 cosTheta_i)) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = expf((0.6931f + ((-1.0f / v) - ((sinTheta_O * sinTheta_i) / v))));
	return (0.5f / v) * (t_0 + ((cosTheta_O * (t_0 * 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
    real(4) :: t_0
    t_0 = exp((0.6931e0 + (((-1.0e0) / v) - ((sintheta_o * sintheta_i) / v))))
    code = (0.5e0 / v) * (t_0 + ((costheta_o * (t_0 * costheta_i)) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = exp(Float32(Float32(0.6931) + Float32(Float32(Float32(-1.0) / v) - Float32(Float32(sinTheta_O * sinTheta_i) / v))))
	return Float32(Float32(Float32(0.5) / v) * Float32(t_0 + Float32(Float32(cosTheta_O * Float32(t_0 * cosTheta_i)) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = exp((single(0.6931) + ((single(-1.0) / v) - ((sinTheta_O * sinTheta_i) / v))));
	tmp = (single(0.5) / v) * (t_0 + ((cosTheta_O * (t_0 * cosTheta_i)) / v));
end

Derivation

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. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.8%

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

   10. sub-neg 99.8%

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

   11. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in cosTheta_O around 0 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 98.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right) \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

   2. pow1/3 98.9%

      \[\leadsto \color{blue}{{\left(\left(e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right) \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right)}^{0.3333333333333333}} \]

5. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{{\left({\left(e^{\left(\frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v} - \frac{1}{v}\right) + 0.6931} \cdot \frac{0.5}{v}\right)}^{3}\right)}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. pow-pow 99.8%

      \[\leadsto \color{blue}{{\left(e^{\left(\frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v} - \frac{1}{v}\right) + 0.6931} \cdot \frac{0.5}{v}\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]

   2. metadata-eval 99.8%

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

   3. add-exp-log 99.8%

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

   4. pow1 99.8%

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

   5. *-commutative 99.8%

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

   6. log-prod 99.8%

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

   7. add-log-exp 99.8%

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

   8. +-commutative 99.8%

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

   9. sub-div 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in cosTheta_O around inf 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 98.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right) \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

   2. pow1/3 98.9%

      \[\leadsto \color{blue}{{\left(\left(e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right) \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right)}^{0.3333333333333333}} \]

5. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{{\left({\left(e^{\left(\frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v} - \frac{1}{v}\right) + 0.6931} \cdot \frac{0.5}{v}\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in sinTheta_O around 0 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 98.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right) \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

   2. pow1/3 98.9%

      \[\leadsto \color{blue}{{\left(\left(e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right) \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right)}^{0.3333333333333333}} \]

5. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{{\left({\left(e^{\left(\frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v} - \frac{1}{v}\right) + 0.6931} \cdot \frac{0.5}{v}\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in cosTheta_O around 0 99.7%

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

   1. associate-*r/ 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in sinTheta_O around 0 99.7%

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

10. Final simplification 99.7%

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

Alternative 5: 14.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta_O \leq 3.499999888929329 \cdot 10^{-22}:\\ \;\;\;\;e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta_O \cdot \frac{cosTheta_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_O 3.499999888929329e-22)
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (exp (* cosTheta_O (/ cosTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_O <= 3.499999888929329e-22f) {
		tmp = expf(((sinTheta_O * -sinTheta_i) / v));
	} else {
		tmp = expf((cosTheta_O * (cosTheta_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 (costheta_o <= 3.499999888929329e-22) then
        tmp = exp(((sintheta_o * -sintheta_i) / v))
    else
        tmp = exp((costheta_o * (costheta_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 (cosTheta_O <= Float32(3.499999888929329e-22))
		tmp = exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v));
	else
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_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 (cosTheta_O <= single(3.499999888929329e-22))
		tmp = exp(((sinTheta_O * -sinTheta_i) / v));
	else
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_O < 3.4999999e-22

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in sinTheta_i around inf 13.5%

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

      1. associate-*r/ 13.5%

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

      2. neg-mul-1 13.5%

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

      3. distribute-rgt-neg-in 13.5%

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

   6. Simplified 13.5%

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

   if 3.4999999e-22 < cosTheta_O

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.0%

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

      1. associate--r+ 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in cosTheta_O around inf 22.4%

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

      1. associate-*r/ 22.4%

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

   9. Simplified 22.4%

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

4. Final simplification 16.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta_O \leq 3.499999888929329 \cdot 10^{-22}:\\ \;\;\;\;e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta_O \cdot \frac{cosTheta_i}{v}}\\ \end{array} \]

Alternative 6: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 98.2%

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

   1. associate--r+ 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in sinTheta_O around 0 98.2%

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

8. Final simplification 98.2%

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

Alternative 7: 98.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 98.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right) \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

   2. pow1/3 98.9%

      \[\leadsto \color{blue}{{\left(\left(e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right) \cdot e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}\right)}^{0.3333333333333333}} \]

5. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{{\left({\left(e^{\left(\frac{cosTheta_O \cdot cosTheta_i - sinTheta_O \cdot sinTheta_i}{v} - \frac{1}{v}\right) + 0.6931} \cdot \frac{0.5}{v}\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Taylor expanded in cosTheta_O around 0 99.7%

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

   1. associate-*r/ 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in sinTheta_O around 0 99.7%

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

10. Taylor expanded in v around 0 98.3%

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

11. Final simplification 98.3%

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

Alternative 8: 13.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 98.2%

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

   1. associate--r+ 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in cosTheta_O around inf 14.2%

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

   1. associate-*r/ 14.2%

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

9. Simplified 14.2%

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

10. Final simplification 14.2%

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

Alternative 9: 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 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 98.2%

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

   1. associate--r+ 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in v around inf 6.4%

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

8. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023310 
(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))))))