HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 19.5s

Alternatives: 13

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 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(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 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(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * (((1.0f / v) / sinhf((-1.0f / v))) * (expf((-sinTheta_O * (sinTheta_i / v))) / (-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 = (costheta_i * costheta_o) * (((1.0e0 / v) / sinh(((-1.0e0) / v))) * (exp((-sintheta_o * (sintheta_i / v))) / ((-2.0e0) * v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

4. Applied rewrites 99.0%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 99.0

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * ((expf((-sinTheta_O * (sinTheta_i / v))) / v) / (sinhf((-1.0f / v)) * (-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 = (costheta_i * costheta_o) * ((exp((-sintheta_o * (sintheta_i / v))) / v) / (sinh(((-1.0e0) / v)) * ((-2.0e0) * v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2024230 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))