HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 11.0s

Alternatives: 8

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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in sinTheta_i around 0

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

8. Applied rewrites 99.8%

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

   1. lift-+.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. log-div N/A

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

   5. metadata-eval N/A

      \[\leadsto e^{\left(\frac{6931}{10000} - \frac{1}{v}\right) + \left(\color{blue}{0} - \log \left(2 \cdot v\right)\right)} \]

   6. associate-+r- N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{6931}{10000} - \frac{1}{v}\right) + 0\right) - \log \left(2 \cdot v\right)}} \]

   7. lower--.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{6931}{10000} - \frac{1}{v}\right) + 0\right) - \log \left(2 \cdot v\right)}} \]

   8. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{6931}{10000} - \frac{1}{v}\right) + 0\right)} - \log \left(2 \cdot v\right)} \]

   9. lower-log.f32 99.8

      \[\leadsto e^{\left(\left(0.6931 - \frac{1}{v}\right) + 0\right) - \color{blue}{\log \left(2 \cdot v\right)}} \]

   10. lift-*.f32 N/A

       \[\leadsto e^{\left(\left(\frac{6931}{10000} - \frac{1}{v}\right) + 0\right) - \log \color{blue}{\left(2 \cdot v\right)}} \]

   11. *-commutative N/A

       \[\leadsto e^{\left(\left(\frac{6931}{10000} - \frac{1}{v}\right) + 0\right) - \log \color{blue}{\left(v \cdot 2\right)}} \]

   12. lower-*.f32 99.8

       \[\leadsto e^{\left(\left(0.6931 - \frac{1}{v}\right) + 0\right) - \log \color{blue}{\left(v \cdot 2\right)}} \]

10. Applied rewrites 99.8%

    \[\leadsto e^{\color{blue}{\left(\left(0.6931 - \frac{1}{v}\right) + 0\right) - \log \left(v \cdot 2\right)}} \]
11. Step-by-step derivation

    1. lift-+.f32 N/A

       \[\leadsto e^{\color{blue}{\left(\left(\frac{6931}{10000} - \frac{1}{v}\right) + 0\right)} - \log \left(v \cdot 2\right)} \]

    2. +-rgt-identity 99.8

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

12. Applied rewrites 99.8%

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

Alternative 2: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   4. lift-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. associate-+l- N/A

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

   7. associate-+r- N/A

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

   8. exp-diff N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 86.1%

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

5. Taylor expanded in v around inf

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

   1. exp-sum N/A

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

   2. rem-exp-log N/A

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

   3. rem-exp-log N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.8

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

7. Applied rewrites 99.8%

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

Alternative 3: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. +-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. lower--.f32 N/A

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

   9. div-add-rev N/A

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

   10. lower-/.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f32 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in sinTheta_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 99.8%

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

Alternative 4: 19.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 2.000000047484456 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 2.000000047484456e-32)
   (exp (/ (* cosTheta_i cosTheta_O) v))
   (exp (* (- sinTheta_O) (/ sinTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 2.000000047484456e-32f) {
		tmp = expf(((cosTheta_i * cosTheta_O) / v));
	} else {
		tmp = expf((-sinTheta_O * (sinTheta_i / v)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 2.000000047484456e-32) then
        tmp = exp(((costheta_i * costheta_o) / v))
    else
        tmp = exp((-sintheta_o * (sintheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(2.000000047484456e-32))
		tmp = exp(Float32(Float32(cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp(Float32(Float32(-sinTheta_O) * Float32(sinTheta_i / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(2.000000047484456e-32))
		tmp = exp(((cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp((-sinTheta_O * (sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 2.00000005e-32

   1. Initial program 99.8%

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

   3. Taylor expanded in cosTheta_i around 0

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

      1. lower--.f32 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 10.6

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

   8. Applied rewrites 10.6%

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

   if 2.00000005e-32 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 100.0%

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

   3. Taylor expanded in cosTheta_i around 0

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

      1. lower--.f32 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in sinTheta_i around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in sinTheta_i around inf

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

       1. mul-1-neg N/A

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

       2. associate-/l* N/A

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

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

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

       4. lower-*.f32 N/A

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

       5. lower-neg.f32 N/A

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

       6. lower-/.f32 45.9

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

   11. Applied rewrites 45.9%

       \[\leadsto e^{\color{blue}{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 19.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 2.000000047484456 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta\_O}{v} \cdot sinTheta\_i}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) 2.000000047484456e-32)
   (exp (/ (* cosTheta_i cosTheta_O) v))
   (exp (* (/ (- sinTheta_O) v) sinTheta_i))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= 2.000000047484456e-32f) {
		tmp = expf(((cosTheta_i * cosTheta_O) / v));
	} else {
		tmp = expf(((-sinTheta_O / v) * sinTheta_i));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((sintheta_i * sintheta_o) <= 2.000000047484456e-32) then
        tmp = exp(((costheta_i * costheta_o) / v))
    else
        tmp = exp(((-sintheta_o / v) * sintheta_i))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(2.000000047484456e-32))
		tmp = exp(Float32(Float32(cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp(Float32(Float32(Float32(-sinTheta_O) / v) * sinTheta_i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(2.000000047484456e-32))
		tmp = exp(((cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp(((-sinTheta_O / v) * sinTheta_i));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 2.00000005e-32

   1. Initial program 99.8%

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

   3. Taylor expanded in cosTheta_i around 0

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

      1. lower--.f32 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in cosTheta_i around inf

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 10.6

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

   8. Applied rewrites 10.6%

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

   if 2.00000005e-32 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 100.0%

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

   3. Taylor expanded in cosTheta_i around 0

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

      1. lower--.f32 N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in sinTheta_i around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in sinTheta_i around inf

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

       1. mul-1-neg N/A

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

       2. associate-/l* N/A

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

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

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

       4. lower-*.f32 N/A

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

       5. lower-neg.f32 N/A

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

       6. lower-/.f32 45.9

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

   11. Applied rewrites 45.9%

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

   13. Applied rewrites 45.9%

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

4. Final simplification 16.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq 2.000000047484456 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta\_O}{v} \cdot sinTheta\_i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 97.7

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

8. Applied rewrites 97.3%

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

9. Taylor expanded in sinTheta_i around 0

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

11. Applied rewrites 98.5%

    \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}} \]
12. Add Preprocessing

Alternative 7: 13.4% accurate, 2.3× speedup

Math

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

FPCore

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

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))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in cosTheta_i around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 11.7

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

8. Applied rewrites 11.7%

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

Alternative 8: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / 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 = (0.5e0 / v) * exp(0.6931e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. +-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. lower--.f32 N/A

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

   9. div-add-rev N/A

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

   10. lower-/.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f32 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in v around inf

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

8. Applied rewrites 4.6%

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

Reproduce

herbie shell --seed 2024337 
(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))))))