HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 14.9s

Alternatives: 12

Speedup: 2.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}\\ e^{\mathsf{fma}\left({t_0}^{2}, t_0, \frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (cbrt (+ 0.6931 (log (/ 0.5 v))))))
   (exp
    (fma
     (pow t_0 2.0)
     t_0
     (-
      (/ cosTheta_i (/ v cosTheta_O))
      (fma sinTheta_O (/ sinTheta_i v) (/ 1.0 v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cbrtf((0.6931f + logf((0.5f / v))));
	return expf(fmaf(powf(t_0, 2.0f), t_0, ((cosTheta_i / (v / cosTheta_O)) - fmaf(sinTheta_O, (sinTheta_i / v), (1.0f / v)))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cbrt(Float32(Float32(0.6931) + log(Float32(Float32(0.5) / v))))
	return exp(fma((t_0 ^ Float32(2.0)), t_0, Float32(Float32(cosTheta_i / Float32(v / cosTheta_O)) - fma(sinTheta_O, Float32(sinTheta_i / v), Float32(Float32(1.0) / v)))))
end

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. add-cube-cbrt 99.8%

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

   3. fma-def 99.8%

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

   4. pow2 99.8%

      \[\leadsto e^{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}\right)}^{2}}, \sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}, \frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} \]

   5. associate-*l/ 99.8%

      \[\leadsto e^{\mathsf{fma}\left({\left(\sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}\right)}^{2}, \sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}, \frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O} + \frac{1}{v}\right)\right)} \]

   6. *-commutative 99.8%

      \[\leadsto e^{\mathsf{fma}\left({\left(\sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}\right)}^{2}, \sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}, \frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} + \frac{1}{v}\right)\right)} \]

   7. fma-def 99.8%

      \[\leadsto e^{\mathsf{fma}\left({\left(\sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}\right)}^{2}, \sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}, \frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \color{blue}{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)}\right)} \]

5. Applied egg-rr 99.8%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}\right)}^{2}, \sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}, \frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)}} \]

6. Final simplification 99.8%

   \[\leadsto e^{\mathsf{fma}\left({\left(\sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}\right)}^{2}, \sqrt[3]{0.6931 + \log \left(\frac{0.5}{v}\right)}, \frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)} \]

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = exp(((0.6931e0 + ((((costheta_i * costheta_o) / v) - ((sintheta_o * sintheta_i) / v)) + ((-1.0e0) / v))) + log((1.0e0 / (v * 2.0e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+ 0.6931 (log (/ 0.5 v)))
   (-
    (/ cosTheta_i (/ v cosTheta_O))
    (+ (/ 1.0 v) (/ (* sinTheta_O sinTheta_i) 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_i / (v / cosTheta_O)) - ((1.0f / v) + ((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 = exp(((0.6931e0 + log((0.5e0 / v))) + ((costheta_i / (v / costheta_o)) - ((1.0e0 / v) + ((sintheta_o * sintheta_i) / 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(cosTheta_i / Float32(v / cosTheta_O)) - Float32(Float32(Float32(1.0) / v) + Float32(Float32(sinTheta_O * sinTheta_i) / 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_i / (v / cosTheta_O)) - ((single(1.0) / v) + ((sinTheta_O * sinTheta_i) / v)))));
end

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * (Float32(exp(1)) ^ Float32(Float32(0.6931) + Float32(Float32(Float32(-1.0) - Float32(sinTheta_O * sinTheta_i)) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (single(2.71828182845904523536) ^ (single(0.6931) + ((single(-1.0) - (sinTheta_O * sinTheta_i)) / v)));
end

Derivation

1. Initial program 99.8%

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

   1. exp-sum 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-/r/ 99.7%

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

   6. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

   1. exp-1-e 99.7%

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

   2. associate-+l+ 99.7%

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

   3. associate-/r/ 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in cosTheta_O around 0 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-neg-in 99.7%

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

   3. sub-neg 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. metadata-eval 99.7%

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

   6. *-commutative 99.7%

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

   7. div-sub 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 cosTheta_i around 0 99.8%

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

   1. exp-diff 99.7%

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

   2. exp-sum 99.7%

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

   3. rem-exp-log 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. fma-udef 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in v around 0 99.7%

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

8. Final simplification 99.7%

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

Alternative 6: 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 99.7%

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

5. Taylor expanded in cosTheta_i around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 7: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((0.6931e0 + (((-1.0e0) - (sintheta_o * sintheta_i)) / 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(Float32(-1.0) - Float32(sinTheta_O * sinTheta_i)) / 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) - (sinTheta_O * sinTheta_i)) / v)));
end

Derivation

1. Initial program 99.8%

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

   1. exp-sum 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. +-commutative 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-/r/ 99.7%

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

   6. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

   1. exp-1-e 99.7%

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

   2. associate-+l+ 99.7%

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

   3. associate-/r/ 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in cosTheta_O around 0 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-neg-in 99.7%

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

   3. sub-neg 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. metadata-eval 99.7%

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

   6. *-commutative 99.7%

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

   7. div-sub 99.7%

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

10. Simplified 99.7%

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

    1. pow-to-exp 99.6%

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

    2. e-exp-1 99.6%

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

    3. add-log-exp 99.7%

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

    4. *-un-lft-identity 99.7%

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

12. Applied egg-rr 99.7%

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

13. Final simplification 99.7%

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

Alternative 8: 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 cosTheta_i around 0 99.8%

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

   1. exp-diff 99.7%

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

   2. exp-sum 99.7%

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

   3. rem-exp-log 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. fma-udef 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in sinTheta_O around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. times-frac 99.7%

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

   3. div-exp 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 9: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 v around 0 98.1%

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

5. Taylor expanded in cosTheta_i around 0 98.1%

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

   1. associate-*r/ 98.1%

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

   2. mul-1-neg 98.1%

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

   3. *-commutative 98.1%

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

   4. distribute-neg-in 98.1%

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

   5. metadata-eval 98.1%

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

   6. *-commutative 98.1%

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

7. Simplified 98.1%

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

8. Taylor expanded in sinTheta_i around inf 98.1%

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

   1. associate-*r/ 98.1%

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

   2. neg-mul-1 98.1%

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

   3. distribute-neg-in 98.1%

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

   4. metadata-eval 98.1%

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

   5. *-commutative 98.1%

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

   6. sub-neg 98.1%

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

10. Simplified 98.1%

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

11. Final simplification 98.1%

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

Alternative 10: 98.0% accurate, 2.1× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 cosTheta_i around 0 99.8%

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

   1. exp-diff 99.7%

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

   2. exp-sum 99.7%

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

   3. rem-exp-log 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. fma-udef 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in sinTheta_O around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. times-frac 99.7%

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

   3. div-exp 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in v around 0 98.3%

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

11. Final simplification 98.3%

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

Alternative 11: 97.9% 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 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 v around 0 98.1%

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

5. Taylor expanded in cosTheta_i around 0 98.1%

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

   1. associate-*r/ 98.1%

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

   2. mul-1-neg 98.1%

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

   3. *-commutative 98.1%

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

   4. distribute-neg-in 98.1%

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

   5. metadata-eval 98.1%

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

   6. *-commutative 98.1%

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

7. Simplified 98.1%

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

8. Taylor expanded in sinTheta_i around 0 98.1%

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

9. Final simplification 98.1%

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

Alternative 12: 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%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 v around 0 98.1%

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

5. Taylor expanded in v around inf 6.4%

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

6. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023174 
(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))))))