HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.8%

Time: 19.4s

Alternatives: 9

Speedup: 2.0×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.7% 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: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-1 - sinTheta\_O \cdot sinTheta\_i}{v} + 0.6931}}{v \cdot 2} \end{array} \]

FPCore

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

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) + 0.6931f)) / (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(((((-1.0e0) - (sintheta_o * sintheta_i)) / v) + 0.6931e0)) / (v * 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 99.6

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

7. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta\_O \cdot \left(-sinTheta\_i\right)} + -1}{v} + 0.6931}}{v \cdot 2} \]
8. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-neg.f32 N/A

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

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

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 99.6

      \[\leadsto \frac{e^{\frac{-1 - \color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} + 0.6931}}{v \cdot 2} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v} + 0.6931}}{v \cdot 2} \]
10. Add Preprocessing

Alternative 2: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.6931 + \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -3.999999911278523 \cdot 10^{+22}:\\ \;\;\;\;\left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(0.5, \frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v}, \frac{cosTheta\_i}{v \cdot cosTheta\_O}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<=
      (+
       (+
        0.6931
        (+
         (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_O sinTheta_i) v))
         (/ -1.0 v)))
       (log (/ 1.0 (* v 2.0))))
      -3.999999911278523e+22)
   (*
    (* cosTheta_O cosTheta_O)
    (fma
     0.5
     (/ (* cosTheta_i cosTheta_i) (* v v))
     (/ cosTheta_i (* v cosTheta_O))))
   (/
    (* cosTheta_i (* cosTheta_i (* 0.5 (* cosTheta_O cosTheta_O))))
    (* v v))))

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_O * sinTheta_i) / v)) + (-1.0f / v))) + logf((1.0f / (v * 2.0f)))) <= -3.999999911278523e+22f) {
		tmp = (cosTheta_O * cosTheta_O) * fmaf(0.5f, ((cosTheta_i * cosTheta_i) / (v * v)), (cosTheta_i / (v * cosTheta_O)));
	} else {
		tmp = (cosTheta_i * (cosTheta_i * (0.5f * (cosTheta_O * cosTheta_O)))) / (v * v);
	}
	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_O * sinTheta_i) / v)) + Float32(Float32(-1.0) / v))) + log(Float32(Float32(1.0) / Float32(v * Float32(2.0))))) <= Float32(-3.999999911278523e+22))
		tmp = Float32(Float32(cosTheta_O * cosTheta_O) * fma(Float32(0.5), Float32(Float32(cosTheta_i * cosTheta_i) / Float32(v * v)), Float32(cosTheta_i / Float32(v * cosTheta_O))));
	else
		tmp = Float32(Float32(cosTheta_i * Float32(cosTheta_i * Float32(Float32(0.5) * Float32(cosTheta_O * cosTheta_O)))) / Float32(v * v));
	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)))) < -3.99999991e22

   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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-/.f32 7.2

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

   8. Applied rewrites 7.2%

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

   9. Taylor expanded in cosTheta_O around inf

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-fma.f32 N/A

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

       5. lower-/.f32 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f32 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f32 N/A

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

       10. lower-/.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 53.3

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

   11. Applied rewrites 4.2%

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

   if -3.99999991e22 < (+.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 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 inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 8.9

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

   5. Applied rewrites 8.9%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-/.f32 7.2

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

   8. Applied rewrites 7.2%

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

   9. Taylor expanded in cosTheta_O around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{cosTheta\_O}^{2} \cdot {cosTheta\_i}^{2}}{{v}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f32 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f32 N/A

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

       10. lower-*.f32 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 80.0

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

   11. Applied rewrites 80.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.6931 + \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -3.999999911278523 \cdot 10^{+22}:\\ \;\;\;\;\left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(0.5, \frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v}, \frac{cosTheta\_i}{v \cdot cosTheta\_O}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)\right)}{v \cdot v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{if}\;\left(0.6931 + \left(\left(t\_0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -3.999999911278523 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)\right)}{v \cdot v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_i cosTheta_O) v)))
   (if (<=
        (+
         (+ 0.6931 (+ (- t_0 (/ (* sinTheta_O sinTheta_i) v)) (/ -1.0 v)))
         (log (/ 1.0 (* v 2.0))))
        -3.999999911278523e+22)
     t_0
     (/
      (* cosTheta_i (* cosTheta_i (* 0.5 (* cosTheta_O cosTheta_O))))
      (* v v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) / v;
	float tmp;
	if (((0.6931f + ((t_0 - ((sinTheta_O * sinTheta_i) / v)) + (-1.0f / v))) + logf((1.0f / (v * 2.0f)))) <= -3.999999911278523e+22f) {
		tmp = t_0;
	} else {
		tmp = (cosTheta_i * (cosTheta_i * (0.5f * (cosTheta_O * cosTheta_O)))) / (v * v);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (costheta_i * costheta_o) / v
    if (((0.6931e0 + ((t_0 - ((sintheta_o * sintheta_i) / v)) + ((-1.0e0) / v))) + log((1.0e0 / (v * 2.0e0)))) <= (-3.999999911278523e+22)) then
        tmp = t_0
    else
        tmp = (costheta_i * (costheta_i * (0.5e0 * (costheta_o * costheta_o)))) / (v * v)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) / v)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(0.6931) + Float32(Float32(t_0 - Float32(Float32(sinTheta_O * sinTheta_i) / v)) + Float32(Float32(-1.0) / v))) + log(Float32(Float32(1.0) / Float32(v * Float32(2.0))))) <= Float32(-3.999999911278523e+22))
		tmp = t_0;
	else
		tmp = Float32(Float32(cosTheta_i * Float32(cosTheta_i * Float32(Float32(0.5) * Float32(cosTheta_O * cosTheta_O)))) / Float32(v * v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_i * cosTheta_O) / v;
	tmp = single(0.0);
	if (((single(0.6931) + ((t_0 - ((sinTheta_O * sinTheta_i) / v)) + (single(-1.0) / v))) + log((single(1.0) / (v * single(2.0))))) <= single(-3.999999911278523e+22))
		tmp = t_0;
	else
		tmp = (cosTheta_i * (cosTheta_i * (single(0.5) * (cosTheta_O * cosTheta_O)))) / (v * v);
	end
	tmp_2 = 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)))) < -3.99999991e22

   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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in cosTheta_i around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 11.5

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

   8. Applied rewrites 11.5%

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

   9. Taylor expanded in cosTheta_O around inf

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

       1. lower-/.f32 N/A

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

       2. lower-*.f32 35.2

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

   11. Applied rewrites 35.2%

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

   if -3.99999991e22 < (+.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 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 inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 8.9

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

   5. Applied rewrites 8.9%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-/.f32 7.2

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

   8. Applied rewrites 7.2%

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

   9. Taylor expanded in cosTheta_O around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{cosTheta\_O}^{2} \cdot {cosTheta\_i}^{2}}{{v}^{2}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f32 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f32 N/A

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

       10. lower-*.f32 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 80.0

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

   11. Applied rewrites 80.0%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.6931 + \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -3.999999911278523 \cdot 10^{+22}:\\ \;\;\;\;\frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)\right)}{v \cdot v}\\ \end{array} \]
5. 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 99.6%

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 99.6

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

7. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta\_O \cdot \left(-sinTheta\_i\right)} + -1}{v} + 0.6931}}{v \cdot 2} \]
8. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-neg.f32 N/A

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

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

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 99.6

      \[\leadsto \frac{e^{\frac{-1 - \color{blue}{sinTheta\_O \cdot sinTheta\_i}}{v} + 0.6931}}{v \cdot 2} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{\color{blue}{-1 - sinTheta\_O \cdot sinTheta\_i}}{v} + 0.6931}}{v \cdot 2} \]

10. Taylor expanded in sinTheta_O around 0

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

    1. *-commutative N/A

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

    2. associate-*l/ N/A

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

    3. associate-/l* N/A

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

    4. lower-*.f32 N/A

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

    5. lower-exp.f32 N/A

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

    6. sub-neg N/A

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

    7. lft-mult-inverse N/A

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

    8. associate-*l/ N/A

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

    9. associate-/r* N/A

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

    10. lower-+.f32 N/A

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

    11. associate-/r* N/A

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

    12. associate-*l/ N/A

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

    13. lft-mult-inverse N/A

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

    14. distribute-neg-frac N/A

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

    15. metadata-eval N/A

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

    16. lower-/.f32 N/A

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

    17. lower-/.f32 99.5

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

12. Applied rewrites 99.5%

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

Alternative 5: 98.1% accurate, 2.4× 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 99.6%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. lower-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. distribute-neg-in N/A

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

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

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

   7. mul-1-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 95.4

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

5. Applied rewrites 94.6%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. div-sub N/A

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

   2. lower-exp.f32 N/A

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

   3. div-sub N/A

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

   4. lower-/.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 94.3

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

8. Applied rewrites 94.3%

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

9. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{e^{\frac{-1}{v}}} \]
10. Step-by-step derivation

    1. metadata-eval N/A

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

    2. distribute-neg-frac N/A

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

    3. lower-exp.f32 N/A

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

    4. distribute-neg-frac N/A

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

    5. metadata-eval N/A

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

    6. lower-/.f32 96.9

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

11. Applied rewrites 96.9%

    \[\leadsto \color{blue}{e^{\frac{-1}{v}}} \]
12. Add Preprocessing

Alternative 6: 18.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \leq -9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;\left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(0.5, \frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v}, \frac{\frac{cosTheta\_i}{v} + \frac{1}{cosTheta\_O}}{cosTheta\_O}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta\_i \cdot cosTheta\_i\right) \cdot \mathsf{fma}\left(0.5, \frac{cosTheta\_O \cdot cosTheta\_O}{v \cdot v}, \frac{cosTheta\_O}{v \cdot cosTheta\_i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_i -9.999999682655225e-22)
   (*
    (* cosTheta_O cosTheta_O)
    (fma
     0.5
     (/ (* cosTheta_i cosTheta_i) (* v v))
     (/ (+ (/ cosTheta_i v) (/ 1.0 cosTheta_O)) cosTheta_O)))
   (*
    (* cosTheta_i cosTheta_i)
    (fma
     0.5
     (/ (* cosTheta_O cosTheta_O) (* v v))
     (/ cosTheta_O (* v cosTheta_i))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_i <= -9.999999682655225e-22f) {
		tmp = (cosTheta_O * cosTheta_O) * fmaf(0.5f, ((cosTheta_i * cosTheta_i) / (v * v)), (((cosTheta_i / v) + (1.0f / cosTheta_O)) / cosTheta_O));
	} else {
		tmp = (cosTheta_i * cosTheta_i) * fmaf(0.5f, ((cosTheta_O * cosTheta_O) / (v * v)), (cosTheta_O / (v * cosTheta_i)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (cosTheta_i <= Float32(-9.999999682655225e-22))
		tmp = Float32(Float32(cosTheta_O * cosTheta_O) * fma(Float32(0.5), Float32(Float32(cosTheta_i * cosTheta_i) / Float32(v * v)), Float32(Float32(Float32(cosTheta_i / v) + Float32(Float32(1.0) / cosTheta_O)) / cosTheta_O)));
	else
		tmp = Float32(Float32(cosTheta_i * cosTheta_i) * fma(Float32(0.5), Float32(Float32(cosTheta_O * cosTheta_O) / Float32(v * v)), Float32(cosTheta_O / Float32(v * cosTheta_i))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_i < -9.9999997e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 15.5

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

   5. Applied rewrites 15.5%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-/.f32 8.0

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

   8. Applied rewrites 8.0%

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

   9. Taylor expanded in cosTheta_O around -inf

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. lower-/.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f32 N/A

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

       11. associate-*r/ N/A

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

       12. lower-/.f32 N/A

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

   11. Applied rewrites 5.1%

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

   if -9.9999997e-22 < cosTheta_i

   1. Initial program 99.5%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 12.6

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

   5. Applied rewrites 12.6%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-/.f32 7.0

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

   8. Applied rewrites 7.0%

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

   9. Taylor expanded in cosTheta_i around inf

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-fma.f32 N/A

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

       5. lower-/.f32 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f32 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f32 N/A

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

       10. lower-/.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 66.6

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

   11. Applied rewrites 25.7%

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

Alternative 7: 34.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \leq -9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;\left(cosTheta\_O \cdot cosTheta\_O\right) \cdot \mathsf{fma}\left(0.5, \frac{cosTheta\_i \cdot cosTheta\_i}{v \cdot v}, \frac{cosTheta\_i}{v \cdot cosTheta\_O}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta\_i \cdot cosTheta\_i\right) \cdot \mathsf{fma}\left(0.5, \frac{cosTheta\_O \cdot cosTheta\_O}{v \cdot v}, \frac{cosTheta\_O}{v \cdot cosTheta\_i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_i -9.999999682655225e-22)
   (*
    (* cosTheta_O cosTheta_O)
    (fma
     0.5
     (/ (* cosTheta_i cosTheta_i) (* v v))
     (/ cosTheta_i (* v cosTheta_O))))
   (*
    (* cosTheta_i cosTheta_i)
    (fma
     0.5
     (/ (* cosTheta_O cosTheta_O) (* v v))
     (/ cosTheta_O (* v cosTheta_i))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_i <= -9.999999682655225e-22f) {
		tmp = (cosTheta_O * cosTheta_O) * fmaf(0.5f, ((cosTheta_i * cosTheta_i) / (v * v)), (cosTheta_i / (v * cosTheta_O)));
	} else {
		tmp = (cosTheta_i * cosTheta_i) * fmaf(0.5f, ((cosTheta_O * cosTheta_O) / (v * v)), (cosTheta_O / (v * cosTheta_i)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (cosTheta_i <= Float32(-9.999999682655225e-22))
		tmp = Float32(Float32(cosTheta_O * cosTheta_O) * fma(Float32(0.5), Float32(Float32(cosTheta_i * cosTheta_i) / Float32(v * v)), Float32(cosTheta_i / Float32(v * cosTheta_O))));
	else
		tmp = Float32(Float32(cosTheta_i * cosTheta_i) * fma(Float32(0.5), Float32(Float32(cosTheta_O * cosTheta_O) / Float32(v * v)), Float32(cosTheta_O / Float32(v * cosTheta_i))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_i < -9.9999997e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 15.5

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

   5. Applied rewrites 15.5%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-/.f32 8.0

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

   8. Applied rewrites 8.0%

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

   9. Taylor expanded in cosTheta_O around inf

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-fma.f32 N/A

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

       5. lower-/.f32 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f32 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f32 N/A

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

       10. lower-/.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 52.7

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

   11. Applied rewrites 18.0%

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

   if -9.9999997e-22 < cosTheta_i

   1. Initial program 99.5%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 12.6

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

   5. Applied rewrites 12.6%

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

   6. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-/.f32 7.0

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

   8. Applied rewrites 7.0%

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

   9. Taylor expanded in cosTheta_i around inf

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-fma.f32 N/A

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

       5. lower-/.f32 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f32 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f32 N/A

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

       10. lower-/.f32 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f32 66.6

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

   11. Applied rewrites 25.7%

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

Alternative 8: 38.1% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot cosTheta\_O}{v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 13.3

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

5. Applied rewrites 13.3%

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

6. Taylor expanded in cosTheta_i around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 11.1

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

8. Applied rewrites 11.1%

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

9. Taylor expanded in cosTheta_O around inf

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

    1. lower-/.f32 N/A

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

    2. lower-*.f32 37.4

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

11. Applied rewrites 37.4%

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

12. Final simplification 37.4%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v} \]
13. Add Preprocessing

Alternative 9: 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 99.6%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 13.3

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

5. Applied rewrites 13.3%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 6.5%

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

Reproduce

herbie shell --seed 2024216 
(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))))))