HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 14.1s

Alternatives: 6

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 / exp((log((2.0e0 * v)) + (1.0e0 / v)))) * exp(0.6931e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-exp.f32 N/A

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

   10. lift-log.f32 N/A

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

   11. lift-/.f32 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-exp.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift--.f32 N/A

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

   5. div-sub N/A

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

   6. lift--.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. sub-div N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. associate--l- N/A

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

6. Applied rewrites 84.3%

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

7. Taylor expanded in v around inf

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

    \[\leadsto \frac{1}{e^{\log \left(2 \cdot v\right) + \frac{1}{v}}} \cdot e^{0.6931} \]
11. Add Preprocessing

Alternative 2: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-exp.f32 N/A

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

   10. lift-log.f32 N/A

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

   11. lift-/.f32 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 99.9

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

7. Applied rewrites 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. associate-+l+ N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.9

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

   10. lift--.f32 N/A

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

   11. lift--.f32 N/A

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

   12. associate--l- N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-fma.f32 N/A

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

   16. lower--.f32 99.9

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-exp.f32 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f32 99.9

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((((-1.0f / v) + 0.6931f) - logf((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(((((-1.0e0) / v) + 0.6931e0) - log((2.0e0 * v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in cosTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. sub-neg N/A

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

   9. lower-+.f32 N/A

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

   19. lower-*.f32 N/A

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in sinTheta_i around 0

   \[\leadsto e^{\left(\frac{-1}{v} + \frac{6931}{10000}\right) - \log \left(2 \cdot v\right)} \]
9. Step-by-step derivation

10. Applied rewrites 99.9%

    \[\leadsto e^{\left(\frac{-1}{v} + 0.6931\right) - \log \left(2 \cdot v\right)} \]
11. Add Preprocessing

Alternative 5: 99.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in cosTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. sub-neg N/A

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

   9. lower-+.f32 N/A

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

   19. lower-*.f32 N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 6: 98.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in cosTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. sub-neg N/A

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

   9. lower-+.f32 N/A

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

   19. lower-*.f32 N/A

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in v around 0

   \[\leadsto e^{-1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 98.8%

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

11. Taylor expanded in sinTheta_i around 0

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

13. Applied rewrites 98.8%

    \[\leadsto e^{\frac{-1}{v}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024236 
(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))))))