HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 11.5s

Alternatives: 10

Speedup: 2.1×

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, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot e^{0.6931 - \frac{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v}}}{v} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   3. exp-sum N/A

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000}}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   6. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   8. div-exp N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   9. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   10. lower--.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   11. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\right)} \]

   12. cancel-sign-sub N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\left(\frac{1}{v} - \left(\mathsf{neg}\left(sinTheta\_O\right)\right) \cdot \frac{sinTheta\_i}{v}\right)}} \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \color{blue}{\left(\mathsf{neg}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)}\right)} \]

   14. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \left(\mathsf{neg}\left(\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)\right)\right)} \]

   15. distribute-frac-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \color{blue}{\frac{\mathsf{neg}\left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)} \]

   16. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \frac{\color{blue}{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}}{v}\right)} \]

   17. div-sub N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 - -1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \]
6. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{v}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \]

   3. lift--.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \]

   4. lift-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \]

   5. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}}{v}} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}}{v}} \]

   7. lower-*.f32 99.7

      \[\leadsto \frac{\color{blue}{0.5 \cdot e^{0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}}}{v} \]

   8. lift-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}}{v}}}{v} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O} + 1}{v}}}{v} \]

   10. lower-fma.f32 99.7

       \[\leadsto \frac{0.5 \cdot e^{0.6931 - \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}}{v}}}{v} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{0.5 \cdot e^{0.6931 - \frac{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, 1\right)}{v}}}{v}} \]
8. Add Preprocessing

Alternative 2: 53.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.6931 + \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -1.9999999556392617 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(0.5 \cdot cosTheta\_O, \frac{cosTheta\_i}{v \cdot v}, \frac{-1}{v}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(cosTheta\_O, \mathsf{fma}\left(-0.5, \frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v}, \frac{cosTheta\_i}{v \cdot cosTheta\_O}\right) \cdot \left(-cosTheta\_O\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<=
      (+
       (+
        0.6931
        (+
         (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
         (/ -1.0 v)))
       (log (/ 1.0 (* v 2.0))))
      -1.9999999556392617e+23)
   (/
    1.0
    (fma
     cosTheta_O
     (* cosTheta_i (fma (* 0.5 cosTheta_O) (/ cosTheta_i (* v v)) (/ -1.0 v)))
     1.0))
   (/
    1.0
    (fma
     cosTheta_O
     (*
      (fma
       -0.5
       (/ (* cosTheta_i cosTheta_i) (* v v))
       (/ cosTheta_i (* v cosTheta_O)))
      (- cosTheta_O))
     1.0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (((0.6931f + ((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) + (-1.0f / v))) + logf((1.0f / (v * 2.0f)))) <= -1.9999999556392617e+23f) {
		tmp = 1.0f / fmaf(cosTheta_O, (cosTheta_i * fmaf((0.5f * cosTheta_O), (cosTheta_i / (v * v)), (-1.0f / v))), 1.0f);
	} else {
		tmp = 1.0f / fmaf(cosTheta_O, (fmaf(-0.5f, ((cosTheta_i * cosTheta_i) / (v * v)), (cosTheta_i / (v * cosTheta_O))) * -cosTheta_O), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(0.6931) + Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) + Float32(Float32(-1.0) / v))) + log(Float32(Float32(1.0) / Float32(v * Float32(2.0))))) <= Float32(-1.9999999556392617e+23))
		tmp = Float32(Float32(1.0) / fma(cosTheta_O, Float32(cosTheta_i * fma(Float32(Float32(0.5) * cosTheta_O), Float32(cosTheta_i / Float32(v * v)), Float32(Float32(-1.0) / v))), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / fma(cosTheta_O, Float32(fma(Float32(-0.5), Float32(Float32(cosTheta_i * cosTheta_i) / Float32(v * v)), Float32(cosTheta_i / Float32(v * cosTheta_O))) * Float32(-cosTheta_O)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 #s(literal 1 binary32) v)) #s(literal 6931/10000 binary32)) (log.f32 (/.f32 #s(literal 1 binary32) (*.f32 #s(literal 2 binary32) v)))) < -1.99999996e23

   1. Initial program 100.0%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{\log \left(v \cdot 2\right) - \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931\right)}}} \]

   5. Taylor expanded in cosTheta_i around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{e^{\frac{\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

      5. mul-1-neg N/A

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

      6. lower-*.f32 N/A

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

      7. mul-1-neg N/A

         \[\leadsto \frac{1}{e^{\frac{cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

      8. lower-neg.f32 24.5

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

   7. Simplified 24.5%

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

   8. Taylor expanded in cosTheta_O around 0

      \[\leadsto \frac{1}{\color{blue}{1 + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{v} + \frac{1}{2} \cdot \frac{cosTheta\_O \cdot {cosTheta\_i}^{2}}{{v}^{2}}\right)}} \]

   10. Simplified 61.4%

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

   11. Taylor expanded in cosTheta_i around 0

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{cosTheta\_i \cdot \left(\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2}} - \frac{1}{v}\right)}, 1\right)} \]
   12. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{cosTheta\_i \cdot \left(\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2}} - \frac{1}{v}\right)}, 1\right)} \]

       2. sub-neg N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2}} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)}, 1\right)} \]

       3. associate-/l* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{{v}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), 1\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{{v}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), 1\right)} \]

       5. distribute-neg-frac N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{{v}^{2}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}\right), 1\right)} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{{v}^{2}} + \frac{\color{blue}{-1}}{v}\right), 1\right)} \]

       7. lower-fma.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot cosTheta\_O, \frac{cosTheta\_i}{{v}^{2}}, \frac{-1}{v}\right)}, 1\right)} \]

       8. lower-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot cosTheta\_O}, \frac{cosTheta\_i}{{v}^{2}}, \frac{-1}{v}\right), 1\right)} \]

       9. lower-/.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(\frac{1}{2} \cdot cosTheta\_O, \color{blue}{\frac{cosTheta\_i}{{v}^{2}}}, \frac{-1}{v}\right), 1\right)} \]

       10. unpow2 N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(\frac{1}{2} \cdot cosTheta\_O, \frac{cosTheta\_i}{\color{blue}{v \cdot v}}, \frac{-1}{v}\right), 1\right)} \]

       11. lower-*.f32 N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(\frac{1}{2} \cdot cosTheta\_O, \frac{cosTheta\_i}{\color{blue}{v \cdot v}}, \frac{-1}{v}\right), 1\right)} \]

       12. lower-/.f32 100.0

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

   13. Simplified 100.0%

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

   if -1.99999996e23 < (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 #s(literal 1 binary32) v)) #s(literal 6931/10000 binary32)) (log.f32 (/.f32 #s(literal 1 binary32) (*.f32 #s(literal 2 binary32) v))))

   1. Initial program 98.8%

      \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{e^{\log \left(v \cdot 2\right) - \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931\right)}}} \]

   5. Taylor expanded in cosTheta_i around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \frac{1}{e^{\frac{\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

      5. mul-1-neg N/A

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

      6. lower-*.f32 N/A

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

      7. mul-1-neg N/A

         \[\leadsto \frac{1}{e^{\frac{cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

      8. lower-neg.f32 9.0

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

   7. Simplified 9.0%

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

   8. Taylor expanded in cosTheta_O around 0

      \[\leadsto \frac{1}{\color{blue}{1 + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{v} + \frac{1}{2} \cdot \frac{cosTheta\_O \cdot {cosTheta\_i}^{2}}{{v}^{2}}\right)}} \]

   10. Simplified 8.4%

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

   11. Taylor expanded in cosTheta_O around inf

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v} + \frac{1}{2} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}}\right)}, 1\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_O \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}} + -1 \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right)}, 1\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{cosTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}}\right) + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right)}, 1\right)} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_O \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}}\right) + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right), 1\right)} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}}\right)\right)} + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right), 1\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_O \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{{cosTheta\_i}^{2}}{{v}^{2}} \cdot \frac{-1}{2}}\right)\right) + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right), 1\right)} \]

       6. metadata-eval N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_O \cdot \left(\mathsf{neg}\left(\frac{{cosTheta\_i}^{2}}{{v}^{2}} \cdot \color{blue}{\left(-1 + \frac{1}{2}\right)}\right)\right) + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right), 1\right)} \]

       7. distribute-rgt-out N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_O \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}}\right)}\right)\right) + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right), 1\right)} \]

       8. distribute-rgt-neg-in N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\left(\mathsf{neg}\left(cosTheta\_O \cdot \left(-1 \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}}\right)\right)\right)} + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right), 1\right)} \]

       9. mul-1-neg N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \left(\mathsf{neg}\left(cosTheta\_O \cdot \left(-1 \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}}\right)\right)\right) + cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{cosTheta\_i}{cosTheta\_O \cdot v}\right)\right)}, 1\right)} \]

       10. distribute-rgt-neg-out N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \left(\mathsf{neg}\left(cosTheta\_O \cdot \left(-1 \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}}\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(cosTheta\_O \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right)\right)}, 1\right)} \]

       11. distribute-neg-in N/A

           \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\mathsf{neg}\left(\left(cosTheta\_O \cdot \left(-1 \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}} + \frac{1}{2} \cdot \frac{{cosTheta\_i}^{2}}{{v}^{2}}\right) + cosTheta\_O \cdot \frac{cosTheta\_i}{cosTheta\_O \cdot v}\right)\right)}, 1\right)} \]

   13. Simplified 22.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{-cosTheta\_O \cdot \mathsf{fma}\left(-0.5, \frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v}, \frac{cosTheta\_i}{v \cdot cosTheta\_O}\right)}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.6931 + \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -1.9999999556392617 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(0.5 \cdot cosTheta\_O, \frac{cosTheta\_i}{v \cdot v}, \frac{-1}{v}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(cosTheta\_O, \mathsf{fma}\left(-0.5, \frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v}, \frac{cosTheta\_i}{v \cdot cosTheta\_O}\right) \cdot \left(-cosTheta\_O\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   3. exp-sum N/A

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000}}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   6. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   8. div-exp N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   9. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   10. lower--.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   11. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\right)} \]

   12. cancel-sign-sub N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\left(\frac{1}{v} - \left(\mathsf{neg}\left(sinTheta\_O\right)\right) \cdot \frac{sinTheta\_i}{v}\right)}} \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \color{blue}{\left(\mathsf{neg}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)}\right)} \]

   14. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \left(\mathsf{neg}\left(\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)\right)\right)} \]

   15. distribute-frac-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \color{blue}{\frac{\mathsf{neg}\left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)} \]

   16. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \frac{\color{blue}{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}}{v}\right)} \]

   17. div-sub N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 - -1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \]

6. Taylor expanded in sinTheta_O around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{6931}{10000} - \frac{1}{v}}}{v}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}}{v}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}}}{v} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{e^{\frac{6931}{10000} - \frac{1}{v}}}}{v} \]

   5. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot e^{\color{blue}{\frac{6931}{10000} - \frac{1}{v}}}}{v} \]

   6. lower-/.f32 99.6

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 99.7% accurate, 2.1× 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 99.3%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   3. exp-sum N/A

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000}}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}} \]

   6. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot \frac{e^{\frac{6931}{10000}}}{e^{\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   8. div-exp N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   9. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   10. lower--.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   11. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\right)} \]

   12. cancel-sign-sub N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\left(\frac{1}{v} - \left(\mathsf{neg}\left(sinTheta\_O\right)\right) \cdot \frac{sinTheta\_i}{v}\right)}} \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \color{blue}{\left(\mathsf{neg}\left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)}\right)} \]

   14. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \left(\mathsf{neg}\left(\color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right)\right)\right)} \]

   15. distribute-frac-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \color{blue}{\frac{\mathsf{neg}\left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)} \]

   16. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \left(\frac{1}{v} - \frac{\color{blue}{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}}{v}\right)} \]

   17. div-sub N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 - -1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}}} \]

6. Taylor expanded in sinTheta_O around 0

   \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \frac{1}{v}}} \]
7. Step-by-step derivation

   1. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \frac{1}{v}}} \]

   2. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} - \frac{1}{v}}} \]

   3. lower-/.f32 99.2

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 5: 46.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(0.5 \cdot cosTheta\_O, \frac{cosTheta\_i}{v \cdot v}, \frac{-1}{v}\right), 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\log \left(v \cdot 2\right) - \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931\right)}}} \]

5. Taylor expanded in cosTheta_i around inf

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

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}}} \]

   2. distribute-neg-frac N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{1}{e^{\frac{\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   5. mul-1-neg N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\frac{cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   8. lower-neg.f32 15.1

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

7. Simplified 15.1%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \frac{1}{\color{blue}{1 + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{v} + \frac{1}{2} \cdot \frac{cosTheta\_O \cdot {cosTheta\_i}^{2}}{{v}^{2}}\right)}} \]

10. Simplified 29.3%

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

11. Taylor expanded in cosTheta_i around 0

    \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{cosTheta\_i \cdot \left(\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2}} - \frac{1}{v}\right)}, 1\right)} \]
12. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{cosTheta\_i \cdot \left(\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2}} - \frac{1}{v}\right)}, 1\right)} \]

    2. sub-neg N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2}} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)}, 1\right)} \]

    3. associate-/l* N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{{v}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), 1\right)} \]

    4. associate-*r* N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{{v}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right), 1\right)} \]

    5. distribute-neg-frac N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{{v}^{2}} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}\right), 1\right)} \]

    6. metadata-eval N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{{v}^{2}} + \frac{\color{blue}{-1}}{v}\right), 1\right)} \]

    7. lower-fma.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot cosTheta\_O, \frac{cosTheta\_i}{{v}^{2}}, \frac{-1}{v}\right)}, 1\right)} \]

    8. lower-*.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot cosTheta\_O}, \frac{cosTheta\_i}{{v}^{2}}, \frac{-1}{v}\right), 1\right)} \]

    9. lower-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(\frac{1}{2} \cdot cosTheta\_O, \color{blue}{\frac{cosTheta\_i}{{v}^{2}}}, \frac{-1}{v}\right), 1\right)} \]

    10. unpow2 N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(\frac{1}{2} \cdot cosTheta\_O, \frac{cosTheta\_i}{\color{blue}{v \cdot v}}, \frac{-1}{v}\right), 1\right)} \]

    11. lower-*.f32 N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i \cdot \mathsf{fma}\left(\frac{1}{2} \cdot cosTheta\_O, \frac{cosTheta\_i}{\color{blue}{v \cdot v}}, \frac{-1}{v}\right), 1\right)} \]

    12. lower-/.f32 44.6

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

13. Simplified 44.6%

    \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{cosTheta\_i \cdot \mathsf{fma}\left(0.5 \cdot cosTheta\_O, \frac{cosTheta\_i}{v \cdot v}, \frac{-1}{v}\right)}, 1\right)} \]
14. Add Preprocessing

Alternative 6: 22.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, -cosTheta\_i, cosTheta\_O \cdot \left(0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{v \cdot v}, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\log \left(v \cdot 2\right) - \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931\right)}}} \]

5. Taylor expanded in cosTheta_i around inf

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

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}}} \]

   2. distribute-neg-frac N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{1}{e^{\frac{\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   5. mul-1-neg N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\frac{cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   8. lower-neg.f32 15.1

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

7. Simplified 15.1%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \frac{1}{\color{blue}{1 + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{v} + \frac{1}{2} \cdot \frac{cosTheta\_O \cdot {cosTheta\_i}^{2}}{{v}^{2}}\right)}} \]

10. Simplified 29.3%

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

11. Taylor expanded in v around 0

    \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\frac{-1 \cdot \left(cosTheta\_i \cdot v\right) + \frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}{{v}^{2}}}, 1\right)} \]
12. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\frac{-1 \cdot \left(cosTheta\_i \cdot v\right) + \frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}{{v}^{2}}}, 1\right)} \]

    2. mul-1-neg N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\color{blue}{\left(\mathsf{neg}\left(cosTheta\_i \cdot v\right)\right)} + \frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}{{v}^{2}}, 1\right)} \]

    3. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\left(\mathsf{neg}\left(\color{blue}{v \cdot cosTheta\_i}\right)\right) + \frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}{{v}^{2}}, 1\right)} \]

    4. distribute-rgt-neg-in N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\color{blue}{v \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)} + \frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}{{v}^{2}}, 1\right)} \]

    5. mul-1-neg N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{v \cdot \color{blue}{\left(-1 \cdot cosTheta\_i\right)} + \frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}{{v}^{2}}, 1\right)} \]

    6. lower-fma.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\color{blue}{\mathsf{fma}\left(v, -1 \cdot cosTheta\_i, \frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)\right)}}{{v}^{2}}, 1\right)} \]

    7. mul-1-neg N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, \color{blue}{\mathsf{neg}\left(cosTheta\_i\right)}, \frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)\right)}{{v}^{2}}, 1\right)} \]

    8. lower-neg.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, \color{blue}{\mathsf{neg}\left(cosTheta\_i\right)}, \frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)\right)}{{v}^{2}}, 1\right)} \]

    9. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, \mathsf{neg}\left(cosTheta\_i\right), \color{blue}{\left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right) \cdot \frac{1}{2}}\right)}{{v}^{2}}, 1\right)} \]

    10. associate-*l* N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, \mathsf{neg}\left(cosTheta\_i\right), \color{blue}{cosTheta\_O \cdot \left({cosTheta\_i}^{2} \cdot \frac{1}{2}\right)}\right)}{{v}^{2}}, 1\right)} \]

    11. lower-*.f32 N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, \mathsf{neg}\left(cosTheta\_i\right), \color{blue}{cosTheta\_O \cdot \left({cosTheta\_i}^{2} \cdot \frac{1}{2}\right)}\right)}{{v}^{2}}, 1\right)} \]

    12. lower-*.f32 N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_O \cdot \color{blue}{\left({cosTheta\_i}^{2} \cdot \frac{1}{2}\right)}\right)}{{v}^{2}}, 1\right)} \]

    13. unpow2 N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_O \cdot \left(\color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)} \cdot \frac{1}{2}\right)\right)}{{v}^{2}}, 1\right)} \]

    14. lower-*.f32 N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_O \cdot \left(\color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)} \cdot \frac{1}{2}\right)\right)}{{v}^{2}}, 1\right)} \]

    15. unpow2 N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, \mathsf{neg}\left(cosTheta\_i\right), cosTheta\_O \cdot \left(\left(cosTheta\_i \cdot cosTheta\_i\right) \cdot \frac{1}{2}\right)\right)}{\color{blue}{v \cdot v}}, 1\right)} \]

    16. lower-*.f32 23.1

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

13. Simplified 23.1%

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

14. Final simplification 23.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\mathsf{fma}\left(v, -cosTheta\_i, cosTheta\_O \cdot \left(0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{v \cdot v}, 1\right)} \]
15. Add Preprocessing

Alternative 7: 18.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_O \cdot \left(0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)}{v \cdot v}, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\log \left(v \cdot 2\right) - \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931\right)}}} \]

5. Taylor expanded in cosTheta_i around inf

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

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}}} \]

   2. distribute-neg-frac N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{1}{e^{\frac{\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   5. mul-1-neg N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\frac{cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   8. lower-neg.f32 15.1

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

7. Simplified 15.1%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \frac{1}{\color{blue}{1 + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{v} + \frac{1}{2} \cdot \frac{cosTheta\_O \cdot {cosTheta\_i}^{2}}{{v}^{2}}\right)}} \]

10. Simplified 29.3%

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

11. Taylor expanded in cosTheta_i around inf

    \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot {cosTheta\_i}^{2}}{{v}^{2}}}, 1\right)} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}{{v}^{2}}}, 1\right)} \]

    2. lower-/.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}{{v}^{2}}}, 1\right)} \]

    3. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\color{blue}{\left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right) \cdot \frac{1}{2}}}{{v}^{2}}, 1\right)} \]

    4. associate-*l* N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\color{blue}{cosTheta\_O \cdot \left({cosTheta\_i}^{2} \cdot \frac{1}{2}\right)}}{{v}^{2}}, 1\right)} \]

    5. lower-*.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{\color{blue}{cosTheta\_O \cdot \left({cosTheta\_i}^{2} \cdot \frac{1}{2}\right)}}{{v}^{2}}, 1\right)} \]

    6. lower-*.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_O \cdot \color{blue}{\left({cosTheta\_i}^{2} \cdot \frac{1}{2}\right)}}{{v}^{2}}, 1\right)} \]

    7. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_O \cdot \left(\color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)} \cdot \frac{1}{2}\right)}{{v}^{2}}, 1\right)} \]

    8. lower-*.f32 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_O \cdot \left(\color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)} \cdot \frac{1}{2}\right)}{{v}^{2}}, 1\right)} \]

    9. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_O \cdot \left(\left(cosTheta\_i \cdot cosTheta\_i\right) \cdot \frac{1}{2}\right)}{\color{blue}{v \cdot v}}, 1\right)} \]

    10. lower-*.f32 18.8

        \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_O \cdot \left(\left(cosTheta\_i \cdot cosTheta\_i\right) \cdot 0.5\right)}{\color{blue}{v \cdot v}}, 1\right)} \]

13. Simplified 18.8%

    \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\frac{cosTheta\_O \cdot \left(\left(cosTheta\_i \cdot cosTheta\_i\right) \cdot 0.5\right)}{v \cdot v}}, 1\right)} \]

14. Final simplification 18.8%

    \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_O \cdot \left(0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)}{v \cdot v}, 1\right)} \]
15. Add Preprocessing

Alternative 8: 9.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot \left(v \cdot v\right)}{cosTheta\_O \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 2.0 (* v v)) (* cosTheta_O (* cosTheta_O (* cosTheta_i cosTheta_i)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (2.0f * (v * v)) / (cosTheta_O * (cosTheta_O * (cosTheta_i * cosTheta_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 = (2.0e0 * (v * v)) / (costheta_o * (costheta_o * (costheta_i * costheta_i)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(2.0) * Float32(v * v)) / Float32(cosTheta_O * Float32(cosTheta_O * Float32(cosTheta_i * cosTheta_i))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(2.0) * (v * v)) / (cosTheta_O * (cosTheta_O * (cosTheta_i * cosTheta_i)));
end

Derivation

1. Initial program 99.3%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\log \left(v \cdot 2\right) - \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931\right)}}} \]

5. Taylor expanded in cosTheta_i around inf

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

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}}} \]

   2. distribute-neg-frac N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{1}{e^{\frac{\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   5. mul-1-neg N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\frac{cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   8. lower-neg.f32 15.1

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

7. Simplified 15.1%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \frac{1}{\color{blue}{1 + cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{v} + \frac{1}{2} \cdot \frac{cosTheta\_O \cdot {cosTheta\_i}^{2}}{{v}^{2}}\right)}} \]

10. Simplified 29.3%

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

11. Taylor expanded in cosTheta_O around inf

    \[\leadsto \color{blue}{2 \cdot \frac{{v}^{2}}{{cosTheta\_O}^{2} \cdot {cosTheta\_i}^{2}}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{2 \cdot {v}^{2}}{{cosTheta\_O}^{2} \cdot {cosTheta\_i}^{2}}} \]

    2. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{2 \cdot {v}^{2}}{{cosTheta\_O}^{2} \cdot {cosTheta\_i}^{2}}} \]

    3. lower-*.f32 N/A

       \[\leadsto \frac{\color{blue}{2 \cdot {v}^{2}}}{{cosTheta\_O}^{2} \cdot {cosTheta\_i}^{2}} \]

    4. unpow2 N/A

       \[\leadsto \frac{2 \cdot \color{blue}{\left(v \cdot v\right)}}{{cosTheta\_O}^{2} \cdot {cosTheta\_i}^{2}} \]

    5. lower-*.f32 N/A

       \[\leadsto \frac{2 \cdot \color{blue}{\left(v \cdot v\right)}}{{cosTheta\_O}^{2} \cdot {cosTheta\_i}^{2}} \]

    6. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(v \cdot v\right)}{\color{blue}{\left(cosTheta\_O \cdot cosTheta\_O\right)} \cdot {cosTheta\_i}^{2}} \]

    7. associate-*l* N/A

       \[\leadsto \frac{2 \cdot \left(v \cdot v\right)}{\color{blue}{cosTheta\_O \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}} \]

    8. lower-*.f32 N/A

       \[\leadsto \frac{2 \cdot \left(v \cdot v\right)}{\color{blue}{cosTheta\_O \cdot \left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}} \]

    9. lower-*.f32 N/A

       \[\leadsto \frac{2 \cdot \left(v \cdot v\right)}{cosTheta\_O \cdot \color{blue}{\left(cosTheta\_O \cdot {cosTheta\_i}^{2}\right)}} \]

    10. unpow2 N/A

        \[\leadsto \frac{2 \cdot \left(v \cdot v\right)}{cosTheta\_O \cdot \left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)}\right)} \]

    11. lower-*.f32 11.2

        \[\leadsto \frac{2 \cdot \left(v \cdot v\right)}{cosTheta\_O \cdot \left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)}\right)} \]

13. Simplified 11.2%

    \[\leadsto \color{blue}{\frac{2 \cdot \left(v \cdot v\right)}{cosTheta\_O \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)}} \]
14. Add Preprocessing

Alternative 9: 6.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_i}{-v}, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\log \left(v \cdot 2\right) - \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931\right)}}} \]

5. Taylor expanded in cosTheta_i around inf

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

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}}} \]

   2. distribute-neg-frac N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{1}{e^{\frac{\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   5. mul-1-neg N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\frac{cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   8. lower-neg.f32 15.1

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

7. Simplified 15.1%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \frac{1}{\color{blue}{1 + -1 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v} + 1}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right)} + 1} \]

   3. associate-/l* N/A

      \[\leadsto \frac{1}{\left(\mathsf{neg}\left(\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\right)\right) + 1} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{1}{\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(\frac{cosTheta\_i}{v}\right)\right)} + 1} \]

   5. mul-1-neg N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\mathsf{neg}\left(\frac{cosTheta\_i}{v}\right)}, 1\right)} \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\frac{cosTheta\_i}{\mathsf{neg}\left(v\right)}}, 1\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \color{blue}{\frac{cosTheta\_i}{\mathsf{neg}\left(v\right)}}, 1\right)} \]

   10. lower-neg.f32 6.7

       \[\leadsto \frac{1}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_i}{\color{blue}{-v}}, 1\right)} \]

10. Simplified 6.7%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_i}{-v}, 1\right)}} \]
11. Add Preprocessing

Alternative 10: 6.4% accurate, 272.0× 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 99.3%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\log \left(v \cdot 2\right) - \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931\right)}}} \]

5. Taylor expanded in cosTheta_i around inf

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

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}}} \]

   2. distribute-neg-frac N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{v}}}} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{1}{e^{\frac{\color{blue}{cosTheta\_O \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   5. mul-1-neg N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

      \[\leadsto \frac{1}{e^{\frac{cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\_i\right)\right)}}{v}}} \]

   8. lower-neg.f32 15.1

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

7. Simplified 15.1%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 6.5%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))))