HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.5%

Time: 14.4s

Alternatives: 8

Speedup: N/A×

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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{0.5}{v}\right)\\ t_1 := 0.6931 + \frac{\left(cosTheta\_O \cdot cosTheta\_i - sinTheta\_O \cdot sinTheta\_i\right) - 1}{v}\\ \frac{e^{\frac{{t\_1}^{2}}{t\_1 - t\_0}}}{e^{\frac{{\log \left(2 \cdot v\right)}^{2}}{\left(0.6931 + \frac{\left(-sinTheta\_O\right) \cdot sinTheta\_i - 1}{v}\right) - t\_0}}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (log (/ 0.5 v)))
        (t_1
         (+
          0.6931
          (/
           (- (- (* cosTheta_O cosTheta_i) (* sinTheta_O sinTheta_i)) 1.0)
           v))))
   (/
    (exp (/ (pow t_1 2.0) (- t_1 t_0)))
    (exp
     (/
      (pow (log (* 2.0 v)) 2.0)
      (- (+ 0.6931 (/ (- (* (- sinTheta_O) sinTheta_i) 1.0) v)) t_0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = logf((0.5f / v));
	float t_1 = 0.6931f + ((((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) - 1.0f) / v);
	return expf((powf(t_1, 2.0f) / (t_1 - t_0))) / expf((powf(logf((2.0f * v)), 2.0f) / ((0.6931f + (((-sinTheta_O * sinTheta_i) - 1.0f) / v)) - t_0)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: t_1
    t_0 = log((0.5e0 / v))
    t_1 = 0.6931e0 + ((((costheta_o * costheta_i) - (sintheta_o * sintheta_i)) - 1.0e0) / v)
    code = exp(((t_1 ** 2.0e0) / (t_1 - t_0))) / exp(((log((2.0e0 * v)) ** 2.0e0) / ((0.6931e0 + (((-sintheta_o * sintheta_i) - 1.0e0) / v)) - t_0)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log(Float32(Float32(0.5) / v))
	t_1 = Float32(Float32(0.6931) + Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i) - Float32(sinTheta_O * sinTheta_i)) - Float32(1.0)) / v))
	return Float32(exp(Float32((t_1 ^ Float32(2.0)) / Float32(t_1 - t_0))) / exp(Float32((log(Float32(Float32(2.0) * v)) ^ Float32(2.0)) / Float32(Float32(Float32(0.6931) + Float32(Float32(Float32(Float32(-sinTheta_O) * sinTheta_i) - Float32(1.0)) / v)) - t_0))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log((single(0.5) / v));
	t_1 = single(0.6931) + ((((cosTheta_O * cosTheta_i) - (sinTheta_O * sinTheta_i)) - single(1.0)) / v);
	tmp = exp(((t_1 ^ single(2.0)) / (t_1 - t_0))) / exp(((log((single(2.0) * v)) ^ single(2.0)) / ((single(0.6931) + (((-sinTheta_O * sinTheta_i) - single(1.0)) / v)) - t_0)));
end

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 99.8

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ (- (* (- sinTheta_O) sinTheta_i) 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 + (((-sinTheta_O * sinTheta_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 = (0.5e0 / v) * exp((0.6931e0 + (((-sintheta_o * sintheta_i) - 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(Float32(Float32(-sinTheta_O) * sinTheta_i) - 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) + (((-sinTheta_O * sinTheta_i) - single(1.0)) / v)));
end

Derivation

1. Initial program 99.7%

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

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.8

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 99.8

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

7. Applied rewrites 99.8%

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

Alternative 3: 98.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ (fma 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 (0.5f / v) * expf((0.6931f + (fmaf(cosTheta_O, cosTheta_i, -1.0f) / v)));
}

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(fma(cosTheta_O, cosTheta_i, Float32(-1.0)) / v))))
end

Derivation

1. Initial program 99.7%

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

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.8

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f32 98.3

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

7. Applied rewrites 98.3%

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

Alternative 4: 12.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \end{array} \]

FPCore

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

C

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

Fortran

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.8

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around inf

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

7. Applied rewrites 4.5%

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

8. Taylor expanded in cosTheta_i around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 13.2

      \[\leadsto \frac{0.5}{v} \cdot e^{\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v}} \]

10. Applied rewrites 13.2%

    \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}} \]
11. Add Preprocessing

Alternative 5: 4.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around -inf

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

   1. associate-+r+ N/A

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

   2. exp-sum N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. rem-exp-log N/A

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

   6. lower-*.f32 N/A

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

   7. exp-sum N/A

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

   8. rem-exp-log N/A

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

   9. lower-*.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 4.5

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

5. Applied rewrites 4.5%

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

7. Applied rewrites 4.5%

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

Alternative 6: 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 99.7%

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

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.8

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around inf

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

7. Applied rewrites 4.5%

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

Alternative 7: 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(0.5) / Float32(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 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 88.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. log-div N/A

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

   6. associate-+l- N/A

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

   7. exp-diff N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower-log.f32 88.4

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 88.4

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 88.4

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

6. Applied rewrites 88.4%

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

7. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-exp.f32 4.5

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

9. Applied rewrites 4.5%

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

Alternative 8: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around -inf

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

   1. associate-+r+ N/A

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

   2. exp-sum N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. rem-exp-log N/A

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

   6. lower-*.f32 N/A

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

   7. exp-sum N/A

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

   8. rem-exp-log N/A

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

   9. lower-*.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 4.5

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

5. Applied rewrites 4.5%

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

7. Applied rewrites 4.5%

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

Reproduce

herbie shell --seed 2024309 
(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))))))