HairBSDF, Mp, lower

Percentage Accurate: 99.5% → 99.5%

Time: 18.5s

Alternatives: 12

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: 99.5% 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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\\ {\left(e^{e^{\mathsf{log1p}\left({t\_0}^{2}\right)} + -1}\right)}^{t\_0} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (cbrt (log (/ 0.5 v)))))
   (*
    (pow (exp (+ (exp (log1p (pow t_0 2.0))) -1.0)) t_0)
    (exp
     (+
      (-
       (* cosTheta_i (/ cosTheta_O v))
       (fma sinTheta_i (/ sinTheta_O v) (/ 1.0 v)))
      0.6931)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cbrtf(logf((0.5f / v)));
	return powf(expf((expf(log1pf(powf(t_0, 2.0f))) + -1.0f)), t_0) * expf((((cosTheta_i * (cosTheta_O / v)) - fmaf(sinTheta_i, (sinTheta_O / v), (1.0f / v))) + 0.6931f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cbrt(log(Float32(Float32(0.5) / v)))
	return Float32((exp(Float32(exp(log1p((t_0 ^ Float32(2.0)))) + Float32(-1.0))) ^ t_0) * exp(Float32(Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) - fma(sinTheta_i, Float32(sinTheta_O / v), Float32(Float32(1.0) / v))) + Float32(0.6931))))
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. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. exp-sum 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/l* 99.6%

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

   9. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. add-exp-log 99.6%

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

   2. add-cube-cbrt 99.7%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

   3. exp-prod 99.7%

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \left(\frac{0.5}{v}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

   4. pow2 99.7%

      \[\leadsto {\left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]
7. Step-by-step derivation

   1. expm1-log1p-u 99.7%

      \[\leadsto {\left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}\right)\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

   2. expm1-undefine 99.7%

      \[\leadsto {\left(e^{\color{blue}{e^{\mathsf{log1p}\left({\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}\right)} - 1}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

8. Applied egg-rr 99.7%

   \[\leadsto {\left(e^{\color{blue}{e^{\mathsf{log1p}\left({\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}\right)} - 1}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

9. Final simplification 99.7%

   \[\leadsto {\left(e^{e^{\mathsf{log1p}\left({\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}\right)} + -1}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]
10. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\\ {\left(e^{{t\_0}^{2}}\right)}^{t\_0} \cdot e^{\left(0.6931 + \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) + \frac{-1}{v}} \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cbrt(log(Float32(Float32(0.5) / v)))
	return Float32((exp((t_0 ^ Float32(2.0))) ^ t_0) * exp(Float32(Float32(Float32(0.6931) + Float32(Float32(cosTheta_i * cosTheta_O) / v)) + Float32(Float32(-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. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. exp-sum 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/l* 99.6%

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

   9. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. add-exp-log 99.6%

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

   2. add-cube-cbrt 99.7%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

   3. exp-prod 99.7%

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \left(\frac{0.5}{v}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

   4. pow2 99.7%

      \[\leadsto {\left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

7. Taylor expanded in sinTheta_i around 0 99.7%

   \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot \color{blue}{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \]

8. Final simplification 99.7%

   \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{\left(0.6931 + \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) + \frac{-1}{v}} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.5}{v}}\\ t\_0 \cdot \left(e^{0.6931 + \left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} + \frac{-1}{v}\right)} \cdot t\_0\right) \end{array} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = sqrtf((0.5f / v));
	return t_0 * (expf((0.6931f + ((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + (-1.0f / v)))) * t_0);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = sqrt((0.5e0 / v))
    code = t_0 * (exp((0.6931e0 + ((((costheta_i * costheta_o) - (sintheta_i * sintheta_o)) / v) + ((-1.0e0) / v)))) * t_0)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = sqrt((single(0.5) / v));
	tmp = t_0 * (exp((single(0.6931) + ((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) + (single(-1.0) / v)))) * t_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. Step-by-step derivation

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 99.6%

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

   2. fma-define 99.6%

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

   3. prod-exp 99.6%

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

   4. add-exp-log 99.6%

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

   5. add-sqr-sqrt 99.7%

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

   6. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \sqrt{\frac{0.5}{v}} \cdot \left(e^{0.6931 + \left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} + \frac{-1}{v}\right)} \cdot \sqrt{\frac{0.5}{v}}\right) \]
8. Add Preprocessing

Alternative 4: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((0.6931f + ((cosTheta_i * cosTheta_O) / v)) + (-1.0f / v))) * (0.5f * expf(-logf(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 + ((costheta_i * costheta_o) / v)) + ((-1.0e0) / v))) * (0.5e0 * exp(-log(v)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(0.6931) + ((cosTheta_i * cosTheta_O) / v)) + (single(-1.0) / v))) * (single(0.5) * exp(-log(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. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. exp-sum 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/l* 99.6%

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

   9. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. add-exp-log 99.6%

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

   2. add-cube-cbrt 99.7%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

   3. exp-prod 99.7%

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \left(\frac{0.5}{v}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

   4. pow2 99.7%

      \[\leadsto {\left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

7. Taylor expanded in v around 0 99.7%

   \[\leadsto \color{blue}{e^{\log 0.5 + -1 \cdot \log v} \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
8. Step-by-step derivation

   1. mul-1-neg 99.7%

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

   2. log-rec 99.7%

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

   3. exp-sum 99.7%

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

   4. rem-exp-log 99.7%

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

   5. log-rec 99.7%

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

   6. associate--l+ 99.7%

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

   7. *-commutative 99.7%

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

   8. +-commutative 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-/l* 99.7%

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

   11. fma-undefine 99.7%

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

   12. +-commutative 99.7%

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

   13. associate-+l- 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in sinTheta_i around 0 99.7%

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

11. Final simplification 99.7%

    \[\leadsto e^{\left(0.6931 + \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) + \frac{-1}{v}} \cdot \left(0.5 \cdot e^{-\log v}\right) \]
12. Add Preprocessing

Alternative 5: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ e^{\left(0.6931 + \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) + \frac{-1}{v}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{v}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (exp (+ (+ 0.6931 (/ (* cosTheta_i cosTheta_O) v)) (/ -1.0 v)))
  (expm1 (log1p (/ 0.5 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((0.6931f + ((cosTheta_i * cosTheta_O) / v)) + (-1.0f / v))) * expm1f(log1pf((0.5f / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(Float32(Float32(Float32(0.6931) + Float32(Float32(cosTheta_i * cosTheta_O) / v)) + Float32(Float32(-1.0) / v))) * expm1(log1p(Float32(Float32(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. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. exp-sum 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/l* 99.6%

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

   9. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. add-exp-log 99.6%

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

   2. add-cube-cbrt 99.7%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

   3. exp-prod 99.7%

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \left(\frac{0.5}{v}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

   4. pow2 99.7%

      \[\leadsto {\left(e^{\color{blue}{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \]

7. Taylor expanded in sinTheta_i around 0 99.7%

   \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot \color{blue}{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}\right)\right)} \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}} \]

   2. expm1-undefine 99.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}\right)} - 1\right)} \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}} \]

   3. pow-exp 99.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{e^{{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)}^{2} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}}\right)} - 1\right) \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}} \]

   4. unpow2 99.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(e^{\color{blue}{\left(\sqrt[3]{\log \left(\frac{0.5}{v}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}\right)} \cdot \sqrt[3]{\log \left(\frac{0.5}{v}\right)}}\right)} - 1\right) \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}} \]

   5. add-cube-cbrt 99.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left(e^{\color{blue}{\log \left(\frac{0.5}{v}\right)}}\right)} - 1\right) \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}} \]

   6. add-exp-log 99.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5}{v}}\right)} - 1\right) \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.5}{v}\right)} - 1\right)} \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}} \]
10. Step-by-step derivation

    1. expm1-define 99.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}} \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{v}\right)\right)} \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}} \]

12. Final simplification 99.7%

    \[\leadsto e^{\left(0.6931 + \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) + \frac{-1}{v}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{v}\right)\right) \]
13. Add Preprocessing

Alternative 6: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp(((single(0.6931) + ((cosTheta_i * cosTheta_O) / v)) + (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. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. exp-sum 99.6%

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

   3. rem-exp-log 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. associate--l- 99.6%

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

   7. associate-/l* 99.6%

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

   8. associate-/l* 99.6%

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

   9. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in sinTheta_i around 0 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot e^{\left(0.6931 + \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) + \frac{-1}{v}} \]
7. Add Preprocessing

Alternative 7: 14.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 8.499999924127398 \cdot 10^{-19}:\\ \;\;\;\;{e}^{\left(\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_O 8.499999924127398e-19)
   (pow E (/ (* sinTheta_i (- sinTheta_O)) v))
   (exp (* cosTheta_O (/ cosTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_O <= 8.499999924127398e-19f) {
		tmp = powf(((float) M_E), ((sinTheta_i * -sinTheta_O) / v));
	} else {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (cosTheta_O <= Float32(8.499999924127398e-19))
		tmp = Float32(exp(1)) ^ Float32(Float32(sinTheta_i * Float32(-sinTheta_O)) / v);
	else
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (cosTheta_O <= single(8.499999924127398e-19))
		tmp = single(2.71828182845904523536) ^ ((sinTheta_i * -sinTheta_O) / v);
	else
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_O < 8.49999992e-19

   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. Step-by-step derivation

      1. associate-+l+ 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/l* 99.6%

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

      4. associate-/l* 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in cosTheta_O around inf 73.6%

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

      1. associate--l+ 73.5%

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

      2. associate-*r/ 73.5%

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

      3. metadata-eval 73.5%

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

   7. Simplified 73.5%

      \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \left(\frac{0.6931}{cosTheta\_O} + \left(\left(\frac{cosTheta\_i}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_O}\right) - \left(\frac{1}{cosTheta\_O \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_O \cdot v}\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 73.5%

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

      2. exp-prod 73.5%

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

      3. exp-1-e 73.5%

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

      4. associate--l+ 73.5%

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

      5. +-commutative 73.5%

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

      6. times-frac 94.2%

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

      7. fma-define 95.2%

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

      8. associate-/r* 95.4%

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

   9. Applied egg-rr 95.4%

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

   10. Taylor expanded in sinTheta_O around inf 12.2%

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

       1. associate-*r/ 12.2%

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

       2. mul-1-neg 12.2%

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

       3. *-commutative 12.2%

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

       4. distribute-lft-neg-in 12.2%

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

   12. Simplified 12.2%

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

   if 8.49999992e-19 < cosTheta_O

   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. Step-by-step derivation

      1. associate-+l+ 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/l* 99.6%

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

      4. associate-/l* 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in cosTheta_O around inf 92.1%

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

      1. associate--l+ 92.1%

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

      2. associate-*r/ 92.1%

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

      3. metadata-eval 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in cosTheta_O around inf 20.1%

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

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 8.499999924127398 \cdot 10^{-19}:\\ \;\;\;\;{e}^{\left(\frac{sinTheta\_i \cdot \left(-sinTheta\_O\right)}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 14.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 8.499999924127398 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_O 8.499999924127398e-19)
   (/ 1.0 (exp (/ (* sinTheta_i sinTheta_O) v)))
   (exp (* cosTheta_O (/ cosTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_O <= 8.499999924127398e-19f) {
		tmp = 1.0f / expf(((sinTheta_i * sinTheta_O) / v));
	} else {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (costheta_o <= 8.499999924127398e-19) then
        tmp = 1.0e0 / exp(((sintheta_i * sintheta_o) / v))
    else
        tmp = exp((costheta_o * (costheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (cosTheta_O <= Float32(8.499999924127398e-19))
		tmp = Float32(Float32(1.0) / exp(Float32(Float32(sinTheta_i * sinTheta_O) / v)));
	else
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (cosTheta_O <= single(8.499999924127398e-19))
		tmp = single(1.0) / exp(((sinTheta_i * sinTheta_O) / v));
	else
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_O < 8.49999992e-19

   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. Step-by-step derivation

      1. associate-+l+ 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/l* 99.6%

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

      4. associate-/l* 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r+ 99.6%

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

      2. fma-define 99.6%

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

      3. prod-exp 99.6%

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

      4. add-exp-log 99.7%

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

      5. associate-*r/ 99.7%

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

      6. *-commutative 99.7%

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

      7. clear-num 99.7%

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

      8. *-un-lft-identity 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. add-exp-log 99.6%

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

      2. associate-/r* 99.6%

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

      3. log-div 99.6%

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

      4. add-log-exp 99.6%

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

      5. +-commutative 99.6%

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

      6. sub-div 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in sinTheta_i around inf 12.2%

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

   if 8.49999992e-19 < cosTheta_O

   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. Step-by-step derivation

      1. associate-+l+ 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/l* 99.6%

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

      4. associate-/l* 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in cosTheta_O around inf 92.1%

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

      1. associate--l+ 92.1%

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

      2. associate-*r/ 92.1%

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

      3. metadata-eval 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in cosTheta_O around inf 20.1%

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

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 8.499999924127398 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ 99.6%

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

   2. fma-define 99.6%

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

   3. prod-exp 99.6%

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

   4. add-exp-log 99.6%

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

   5. associate-*r/ 99.6%

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

   6. *-commutative 99.6%

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

   7. clear-num 99.5%

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

   8. *-un-lft-identity 99.5%

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

6. Applied egg-rr 99.5%

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

   1. div-inv 99.5%

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

   2. +-commutative 99.5%

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

   3. sub-div 99.5%

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

8. Applied egg-rr 99.5%

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

   1. associate-/r* 99.5%

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

   2. metadata-eval 99.5%

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

   3. sub-neg 99.5%

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. metadata-eval 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in cosTheta_O around inf 99.5%

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

12. Taylor expanded in cosTheta_O around 0 99.6%

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

    1. associate-*r/ 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 10: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((cosTheta_O * (cosTheta_i - (1.0f / 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 = exp(((costheta_o * (costheta_i - (1.0e0 / costheta_o))) / v))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((cosTheta_O * (cosTheta_i - (single(1.0) / 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. Step-by-step derivation

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_O around inf 78.5%

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

   1. associate--l+ 78.4%

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

   2. associate-*r/ 78.4%

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

   3. metadata-eval 78.4%

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

7. Simplified 78.4%

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

8. Taylor expanded in sinTheta_O around 0 96.4%

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

   1. associate--l+ 96.4%

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

   2. associate-*r/ 96.4%

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

   3. metadata-eval 96.4%

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

   4. *-commutative 96.4%

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

10. Simplified 96.4%

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

11. Taylor expanded in v around 0 97.0%

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

Alternative 11: 13.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_O around inf 78.5%

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

   1. associate--l+ 78.4%

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

   2. associate-*r/ 78.4%

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

   3. metadata-eval 78.4%

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

7. Simplified 78.4%

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

8. Taylor expanded in cosTheta_O around inf 13.2%

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

Alternative 12: 6.4% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l+ 99.6%

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

   2. associate--l- 99.6%

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

   3. associate-/l* 99.6%

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

   4. associate-/l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_O around inf 78.5%

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

   1. associate--l+ 78.4%

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

   2. associate-*r/ 78.4%

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

   3. metadata-eval 78.4%

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

7. Simplified 78.4%

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

8. Taylor expanded in cosTheta_O around inf 13.2%

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

9. Taylor expanded in cosTheta_O around 0 6.5%

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

Reproduce

herbie shell --seed 2024146 
(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))))))