HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 14.8s

Alternatives: 11

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, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{e^{0.6931 - \frac{1}{v}} \cdot \frac{0.5}{v}}\right)}^{3} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow (cbrt (* (exp (- 0.6931 (/ 1.0 v))) (/ 0.5 v))) 3.0))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(cbrtf((expf((0.6931f - (1.0f / v))) * (0.5f / v))), 3.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return cbrt(Float32(exp(Float32(Float32(0.6931) - Float32(Float32(1.0) / v))) * Float32(Float32(0.5) / v))) ^ Float32(3.0)
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. Step-by-step derivation

   1. add-cube-cbrt 99.7%

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

   2. pow3 99.7%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in sinTheta_O around 0 99.8%

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

7. Taylor expanded in cosTheta_O around 0 99.8%

   \[\leadsto {\left(\sqrt[3]{e^{\color{blue}{0.6931} - \frac{1}{v}} \cdot \frac{0.5}{v}}\right)}^{3} \]

8. Final simplification 99.8%

   \[\leadsto {\left(\sqrt[3]{e^{0.6931 - \frac{1}{v}} \cdot \frac{0.5}{v}}\right)}^{3} \]

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+ (+ 0.6931 (log (/ 0.5 v))) (* cosTheta_O (/ cosTheta_i 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))) + (cosTheta_O * (cosTheta_i / 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))) + (costheta_o * (costheta_i / v))) + ((-1.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(0.6931) + log(Float32(Float32(0.5) / v))) + Float32(cosTheta_O * Float32(cosTheta_i / 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))) + (cosTheta_O * (cosTheta_i / v))) + (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 sinTheta_i around 0 99.8%

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

   1. associate-+r+ 99.8%

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

   2. associate-*r/ 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((single(0.6931) - (single(1.0) / v))) * (single(0.5) / 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 99.7%

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

   2. *-commutative 99.7%

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

   3. rem-exp-log 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. +-rgt-identity 99.7%

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

   7. metadata-eval 99.7%

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

   8. metadata-eval 99.7%

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

   9. +-rgt-identity 99.7%

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

   10. sub-neg 99.7%

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

   11. associate-+l+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in sinTheta_O around 0 99.7%

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

5. Taylor expanded in cosTheta_O around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 71.2% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ 1 + \left(-1 + sinTheta_i \cdot \frac{sinTheta_O}{v}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) + (single(-1.0) + (sinTheta_i * (sinTheta_O / 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 sinTheta_i around inf 10.5%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 42.2%

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

   1. mul-1-neg 42.2%

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

   2. associate-*l/ 21.1%

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

   3. distribute-rgt-neg-in 21.1%

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

10. Simplified 21.1%

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

    1. expm1-log1p-u 20.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)\right)\right)} \]

    2. expm1-udef 71.2%

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

12. Applied egg-rr 71.6%

    \[\leadsto \color{blue}{\left(1 + \frac{sinTheta_O}{\frac{v}{sinTheta_i}}\right) - 1} \]
13. Step-by-step derivation

    1. associate--l+ 71.6%

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

    2. sub-neg 71.6%

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

    3. metadata-eval 71.6%

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

    4. +-commutative 71.6%

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

    5. associate-/r/ 71.6%

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

14. Simplified 71.6%

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

15. Final simplification 71.6%

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

Alternative 5: 71.1% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \left(1 + \frac{sinTheta_O}{\frac{v}{sinTheta_i}}\right) + -1 \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(1.0) + (sinTheta_O / (v / sinTheta_i))) + 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 sinTheta_i around inf 10.5%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 42.2%

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

   1. mul-1-neg 42.2%

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

   2. associate-*l/ 21.1%

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

   3. distribute-rgt-neg-in 21.1%

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

10. Simplified 21.1%

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

    1. expm1-log1p-u 20.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta_O}{v} \cdot \left(-sinTheta_i\right)\right)\right)} \]

    2. expm1-udef 71.2%

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

12. Applied egg-rr 71.6%

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

13. Final simplification 71.6%

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

Alternative 6: 38.6% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{v} \cdot \left(sinTheta_O \cdot sinTheta_i\right) \end{array} \]

FPCore

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

C

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

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 / v) * (sintheta_o * sintheta_i)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(1.0) / v) * (sinTheta_O * sinTheta_i);
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 sinTheta_i around inf 10.5%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 42.2%

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

   1. mul-1-neg 42.2%

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

   2. associate-*l/ 21.1%

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

   3. distribute-rgt-neg-in 21.1%

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

10. Simplified 21.1%

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

    1. associate-*l/ 42.2%

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

    2. *-un-lft-identity 42.2%

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

    3. times-frac 21.1%

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

    4. add-sqr-sqrt 10.2%

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

    5. sqrt-unprod 47.1%

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

    6. sqr-neg 47.1%

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

    7. sqrt-unprod 10.9%

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

    8. add-sqr-sqrt 21.1%

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

    9. times-frac 42.2%

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

    10. *-un-lft-identity 42.2%

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

    11. clear-num 42.2%

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

12. Applied egg-rr 42.2%

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

    1. associate-/r/ 42.2%

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

14. Simplified 42.2%

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

15. Final simplification 42.2%

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

Alternative 7: 39.0% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{sinTheta_O \cdot sinTheta_i}} \end{array} \]

FPCore

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

C

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

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 / (v / (sintheta_o * sintheta_i))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (v / (sinTheta_O * sinTheta_i));
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 sinTheta_i around inf 10.5%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 42.2%

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

   1. mul-1-neg 42.2%

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

   2. associate-*l/ 21.1%

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

   3. distribute-rgt-neg-in 21.1%

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

10. Simplified 21.1%

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

    1. associate-*l/ 42.2%

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

    2. *-un-lft-identity 42.2%

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

    3. times-frac 21.1%

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

    4. add-sqr-sqrt 10.2%

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

    5. sqrt-unprod 47.1%

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

    6. sqr-neg 47.1%

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

    7. sqrt-unprod 10.9%

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

    8. add-sqr-sqrt 21.1%

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

    9. times-frac 42.2%

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

    10. *-un-lft-identity 42.2%

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

    11. clear-num 42.2%

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

12. Applied egg-rr 42.2%

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

13. Final simplification 42.2%

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

Alternative 8: 38.7% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O \cdot sinTheta_i}{-v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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 sinTheta_i around inf 10.5%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 42.2%

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

   1. mul-1-neg 42.2%

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

   2. associate-*l/ 21.1%

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

   3. distribute-rgt-neg-in 21.1%

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

10. Simplified 21.1%

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

    1. associate-*l/ 42.2%

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

    2. *-un-lft-identity 42.2%

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

    3. times-frac 21.1%

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

    4. add-sqr-sqrt 10.2%

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

    5. sqrt-unprod 47.1%

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

    6. sqr-neg 47.1%

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

    7. sqrt-unprod 10.9%

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

    8. add-sqr-sqrt 21.1%

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

    9. times-frac 42.2%

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

    10. *-un-lft-identity 42.2%

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

    11. frac-2neg 42.2%

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

    12. distribute-rgt-neg-in 42.2%

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

    13. add-sqr-sqrt 18.7%

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

    14. sqrt-unprod 53.0%

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

    15. sqr-neg 53.0%

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

    16. sqrt-unprod 23.5%

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

    17. add-sqr-sqrt 42.2%

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

12. Applied egg-rr 42.2%

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

13. Final simplification 42.2%

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

Alternative 9: 20.2% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ sinTheta_O \cdot \frac{sinTheta_i}{v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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 sinTheta_i around inf 10.5%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 42.2%

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

   1. mul-1-neg 42.2%

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

   2. associate-*l/ 21.1%

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

   3. distribute-rgt-neg-in 21.1%

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

10. Simplified 21.1%

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

    1. *-commutative 21.1%

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

    2. clear-num 21.1%

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

    3. un-div-inv 21.1%

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

    4. add-sqr-sqrt 10.2%

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

    5. sqrt-unprod 47.1%

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

    6. sqr-neg 47.1%

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

    7. sqrt-unprod 10.9%

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

    8. add-sqr-sqrt 21.1%

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

12. Applied egg-rr 21.1%

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

    1. associate-/r/ 21.1%

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

14. Simplified 21.1%

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

15. Final simplification 21.1%

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

Alternative 10: 20.2% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O}{\frac{v}{sinTheta_i}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sinTheta_O / (v / sinTheta_i);
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 sinTheta_i around inf 10.5%

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

5. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. associate-/l* 6.2%

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

   2. neg-mul-1 6.2%

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

   3. unsub-neg 6.2%

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

   4. associate-/r/ 6.2%

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

7. Simplified 6.2%

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

8. Taylor expanded in sinTheta_O around inf 42.2%

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

   1. mul-1-neg 42.2%

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

   2. associate-*l/ 21.1%

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

   3. distribute-rgt-neg-in 21.1%

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

10. Simplified 21.1%

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

    1. associate-*l/ 42.2%

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

    2. *-un-lft-identity 42.2%

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

    3. times-frac 21.1%

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

    4. add-sqr-sqrt 10.2%

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

    5. sqrt-unprod 47.1%

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

    6. sqr-neg 47.1%

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

    7. sqrt-unprod 10.9%

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

    8. add-sqr-sqrt 21.1%

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

    9. times-frac 42.2%

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

    10. *-un-lft-identity 42.2%

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

    11. associate-/l* 21.1%

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

12. Applied egg-rr 21.1%

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

13. Final simplification 21.1%

    \[\leadsto \frac{sinTheta_O}{\frac{v}{sinTheta_i}} \]

Alternative 11: 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 sinTheta_i around inf 10.5%

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

5. Taylor expanded in sinTheta_O around 0 6.4%

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

6. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023298 
(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))))))