HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 18.4s

Alternatives: 8

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}\\ \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot e^{\left(\log \left(v \cdot 2\right) - t\_0\right) \cdot 0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{t\_0}}\right)}^{2}}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 0.6931 (/ (fma cosTheta_O cosTheta_i -1.0) v))))
   (/
    1.0
    (/
     (*
      (pow (cbrt v) 2.0)
      (exp (* (- (log (* v 2.0)) t_0) 0.3333333333333333)))
     (pow (cbrt (* 0.5 (exp t_0))) 2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 0.6931f + (fmaf(cosTheta_O, cosTheta_i, -1.0f) / v);
	return 1.0f / ((powf(cbrtf(v), 2.0f) * expf(((logf((v * 2.0f)) - t_0) * 0.3333333333333333f))) / powf(cbrtf((0.5f * expf(t_0))), 2.0f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(0.6931) + Float32(fma(cosTheta_O, cosTheta_i, Float32(-1.0)) / v))
	return Float32(Float32(1.0) / Float32(Float32((cbrt(v) ^ Float32(2.0)) * exp(Float32(Float32(log(Float32(v * Float32(2.0))) - t_0) * Float32(0.3333333333333333)))) / (cbrt(Float32(Float32(0.5) * exp(t_0))) ^ Float32(2.0))))
end

Derivation

1. Initial program 99.4%

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

   1. associate-+l+ 99.4%

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

   2. associate--l- 99.4%

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

   3. associate-/l* 99.4%

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

   4. associate-/l* 99.4%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. add-cube-cbrt 99.8%

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

   2. add-cube-cbrt 99.8%

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

   3. times-frac 99.8%

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

9. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.8%

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

11. Simplified 99.8%

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

    1. pow1/3 99.8%

       \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot \color{blue}{{\left(2 \cdot \frac{v}{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{0.3333333333333333}}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    2. associate-*r/ 99.8%

       \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\color{blue}{\left(\frac{2 \cdot v}{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    3. *-commutative 99.8%

       \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\left(\frac{\color{blue}{v \cdot 2}}{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    4. metadata-eval 99.8%

       \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\left(\frac{v \cdot \color{blue}{\frac{1}{0.5}}}{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    5. div-inv 99.8%

       \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\left(\frac{\color{blue}{\frac{v}{0.5}}}{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    6. add-exp-log 99.8%

       \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\left(\frac{\color{blue}{e^{\log \left(\frac{v}{0.5}\right)}}}{e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    7. exp-sum 99.8%

       \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\left(\frac{e^{\log \left(\frac{v}{0.5}\right)}}{\color{blue}{e^{0.6931} \cdot e^{\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}}\right)}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    8. metadata-eval 99.8%

       \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\left(\frac{e^{\log \left(\frac{v}{0.5}\right)}}{e^{0.6931} \cdot e^{\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, \color{blue}{-1}\right)}{v}}}\right)}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    9. fma-neg 99.8%

       \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\left(\frac{e^{\log \left(\frac{v}{0.5}\right)}}{e^{0.6931} \cdot e^{\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i - 1}}{v}}}\right)}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    10. prod-exp 99.8%

        \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\left(\frac{e^{\log \left(\frac{v}{0.5}\right)}}{\color{blue}{e^{0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}}}\right)}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    11. exp-diff 99.8%

        \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot {\color{blue}{\left(e^{\log \left(\frac{v}{0.5}\right) - \left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}\right)}\right)}}^{0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

    12. pow-to-exp 99.8%

        \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot \color{blue}{e^{\log \left(e^{\log \left(\frac{v}{0.5}\right) - \left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}\right)}\right) \cdot 0.3333333333333333}}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

13. Applied egg-rr 99.8%

    \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot \color{blue}{e^{\left(\log \left(2 \cdot v\right) - \left(0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}\right)\right) \cdot 0.3333333333333333}}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]

14. Final simplification 99.8%

    \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot e^{\left(\log \left(v \cdot 2\right) - \left(0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}\right)\right) \cdot 0.3333333333333333}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]
15. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. associate-+l+ 99.4%

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

   2. associate--l- 99.4%

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

   3. associate-/l* 99.4%

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

   4. associate-/l* 99.4%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. add-cube-cbrt 99.8%

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

   2. add-cube-cbrt 99.8%

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

   3. times-frac 99.8%

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

9. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.8%

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

11. Simplified 99.8%

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

12. Taylor expanded in cosTheta_O around 0 99.8%

    \[\leadsto \frac{1}{\frac{{\left(\sqrt[3]{v}\right)}^{2} \cdot \color{blue}{\left({\left(\frac{1 \cdot v}{e^{0.6931 - \frac{1}{v}}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{2}\right)}}{{\left(\sqrt[3]{0.5 \cdot e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}\right)}^{2}}} \]
13. Step-by-step derivation

    1. unpow1/3 99.8%

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

    2. *-lft-identity 99.8%

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

14. Simplified 99.8%

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

15. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. associate-+l+ 99.4%

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

   2. associate--l- 99.4%

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

   3. associate-/l* 99.4%

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

   4. associate-/l* 99.4%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. add-cube-cbrt 99.8%

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

   2. add-cube-cbrt 99.8%

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

   3. times-frac 99.8%

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

9. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.8%

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

11. Simplified 99.8%

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

12. Taylor expanded in cosTheta_O around 0 99.8%

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

13. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. associate-+l+ 99.4%

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

   2. associate--l- 99.4%

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

   3. associate-/l* 99.4%

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

   4. associate-/l* 99.4%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in sinTheta_i around 0 99.8%

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

8. Taylor expanded in cosTheta_O around 0 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

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(0.5) * Float32(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 99.4%

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

   1. +-commutative 99.4%

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

   2. exp-sum 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.4%

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

   10. +-commutative 99.4%

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

   11. associate--l- 99.4%

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

   12. associate-+r- 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

7. Final simplification 99.8%

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

Alternative 6: 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 99.4%

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

   1. associate-+l+ 99.4%

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

   2. associate--l- 99.4%

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

   3. associate-/l* 99.4%

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

   4. associate-/l* 99.4%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.4%

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

6. Taylor expanded in cosTheta_O around 0 99.4%

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

   1. sub-neg 99.4%

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

   2. exp-sum 99.4%

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

   3. exp-sum 99.4%

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

   4. rem-exp-log 99.4%

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

   5. distribute-neg-frac 99.4%

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

   6. metadata-eval 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in v around 0 99.8%

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

    1. associate-*r/ 99.8%

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

    2. prod-exp 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 7: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. exp-sum 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.4%

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

   10. +-commutative 99.4%

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

   11. associate--l- 99.4%

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

   12. associate-+r- 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.7%

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

   3. exp-1-e 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-/l* 99.7%

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

   6. fma-define 99.7%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in cosTheta_O around 0 99.7%

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

9. Taylor expanded in v around 0 98.3%

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

10. Final simplification 98.3%

    \[\leadsto 0.5 \cdot \frac{{e}^{\left(\frac{-1}{v}\right)}}{v} \]
11. Add Preprocessing

Alternative 8: 4.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (expf(0.6931f) / 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) / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. exp-sum 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.4%

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

   10. +-commutative 99.4%

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

   11. associate--l- 99.4%

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

   12. associate-+r- 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around inf 4.6%

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

6. Final simplification 4.6%

   \[\leadsto 0.5 \cdot \frac{e^{0.6931}}{v} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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))))))