HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 14.2s

Alternatives: 9

Speedup: 1.0×

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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \cdot \left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot \frac{1}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(\left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. *-commutative 98.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(\color{blue}{\left(cosTheta_O \cdot cosTheta_i\right)} \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Final simplification 98.6%

   \[\leadsto \frac{e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \cdot \left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot \frac{1}{v}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.2%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.2%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.2%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.3%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]

4. Taylor expanded in sinTheta_i around 0 98.2%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v} \]
5. Step-by-step derivation

   1. associate-/l/ 98.1%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \color{blue}{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

   2. *-un-lft-identity 98.1%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \frac{\color{blue}{1 \cdot cosTheta_i}}{v \cdot \frac{v}{cosTheta_O}} \]

   3. times-frac 98.5%

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

   4. associate-/r/ 98.4%

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

6. Applied egg-rr 98.4%

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

   1. sub-neg 98.4%

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

   2. rec-exp 98.4%

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

8. Applied egg-rr 98.4%

   \[\leadsto \frac{1}{\color{blue}{e^{\frac{1}{v}} + \left(-e^{-\frac{1}{v}}\right)}} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]
9. Step-by-step derivation

   1. sub-neg 98.4%

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

   2. distribute-neg-frac 98.4%

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

   3. metadata-eval 98.4%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

    \[\leadsto \frac{1}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)\right) \]

Alternative 3: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{v} \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.2%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.2%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.2%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.3%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]

4. Taylor expanded in sinTheta_i around 0 98.2%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v} \]
5. Step-by-step derivation

   1. associate-/l/ 98.1%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \color{blue}{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

   2. *-un-lft-identity 98.1%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \frac{\color{blue}{1 \cdot cosTheta_i}}{v \cdot \frac{v}{cosTheta_O}} \]

   3. times-frac 98.5%

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

   4. associate-/r/ 98.4%

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

6. Applied egg-rr 98.4%

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

   1. rec-exp 98.4%

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

   2. sinh-undef 98.4%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]

   3. *-commutative 98.4%

      \[\leadsto \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]

8. Applied egg-rr 98.4%

   \[\leadsto \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]

9. Final simplification 98.4%

   \[\leadsto \left(\frac{1}{v} \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (1.0f / (sinhf((1.0f / v)) * 2.0f)) * ((cosTheta_i * (cosTheta_O / v)) / v);
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.2%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.2%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.2%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.3%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]

4. Taylor expanded in sinTheta_i around 0 98.2%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v} \]
5. Step-by-step derivation

   1. associate-/l/ 98.1%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \color{blue}{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}} \]

   2. *-un-lft-identity 98.1%

      \[\leadsto \frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \frac{\color{blue}{1 \cdot cosTheta_i}}{v \cdot \frac{v}{cosTheta_O}} \]

   3. times-frac 98.5%

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

   4. associate-/r/ 98.4%

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

6. Applied egg-rr 98.4%

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

   1. rec-exp 98.4%

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

   2. sinh-undef 98.4%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]

   3. *-commutative 98.4%

      \[\leadsto \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]

8. Applied egg-rr 98.4%

   \[\leadsto \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]
9. Step-by-step derivation

   1. associate-*l/ 98.2%

      \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\frac{1 \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)}{v}} \]

   2. *-un-lft-identity 98.2%

      \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O}}{v} \]

   3. associate-*l/ 98.1%

      \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}}}{v} \]

   4. div-inv 98.3%

      \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{1}{v}}}{v} \]

   5. associate-*r* 98.3%

      \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{cosTheta_i \cdot \left(cosTheta_O \cdot \frac{1}{v}\right)}}{v} \]

   6. div-inv 98.2%

      \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}}{v} \]

10. Applied egg-rr 98.2%

    \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \color{blue}{\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v}} \]

11. Final simplification 98.2%

    \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{v} \]

Alternative 5: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. times-frac 98.2%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v}} \]

   2. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{\frac{1}{e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   4. exp-neg 98.2%

      \[\leadsto \frac{\color{blue}{e^{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   5. distribute-neg-frac 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{-sinTheta_O \cdot sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   6. *-commutative 98.2%

      \[\leadsto \frac{e^{\frac{-\color{blue}{sinTheta_i \cdot sinTheta_O}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   7. distribute-rgt-neg-out 98.2%

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   8. associate-/l* 98.2%

      \[\leadsto \frac{e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{v} \]

   9. associate-/l* 98.3%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}}{v} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v}} \]

4. Taylor expanded in sinTheta_i around 0 98.2%

   \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v} \]
5. Step-by-step derivation

   1. rec-exp 98.4%

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

   2. sinh-undef 98.4%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]

   3. *-commutative 98.4%

      \[\leadsto \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \cdot \left(\frac{1}{v} \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \frac{1}{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v} \]

7. Final simplification 98.2%

   \[\leadsto \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot 2} \cdot \frac{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}{v} \]

Alternative 6: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* 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 * (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 * (costheta_o * (costheta_i / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(\left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. *-commutative 98.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(\color{blue}{\left(cosTheta_O \cdot cosTheta_i\right)} \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Taylor expanded in v around inf 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. *-commutative 53.5%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} \]

   2. associate-*l/ 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)} \]

   3. *-commutative 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]

6. Simplified 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]

7. Final simplification 53.5%

   \[\leadsto 0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \]

Alternative 7: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot 0.5 \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * (cosTheta_O / v)) * 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 = (costheta_i * (costheta_o / v)) * 0.5e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}\right)}} \]

4. Taylor expanded in v around inf 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. associate-*l/ 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \]

   2. *-commutative 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \]

6. Simplified 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \]

7. Final simplification 53.5%

   \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot 0.5 \]

Alternative 8: 58.1% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (* 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 * ((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 * ((costheta_o * costheta_i) / v)
end function

Julia

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Simplified 98.3%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}\right)}} \]

4. Taylor expanded in v around inf 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]

5. Final simplification 53.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v} \]

Alternative 9: 58.5% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{v}{cosTheta_O \cdot cosTheta_i}} \end{array} \]

FPCore

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

C

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

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 / (costheta_o * costheta_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. div-inv 98.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(\left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. *-commutative 98.6%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(\color{blue}{\left(cosTheta_O \cdot cosTheta_i\right)} \cdot \frac{1}{v}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

3. Applied egg-rr 98.6%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \color{blue}{\left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot \frac{1}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Taylor expanded in v around inf 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v}} \]
5. Step-by-step derivation

   1. *-commutative 53.5%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{v} \]

   2. associate-*l/ 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)} \]

   3. *-commutative 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]

6. Simplified 53.5%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]
7. Step-by-step derivation

   1. *-commutative 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)} \]

   2. associate-*l/ 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} \]

   3. div-inv 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(cosTheta_i \cdot cosTheta_O\right) \cdot \frac{1}{v}\right)} \]

   4. associate-*r* 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta_i \cdot \left(cosTheta_O \cdot \frac{1}{v}\right)\right)} \]

   5. div-inv 53.5%

      \[\leadsto 0.5 \cdot \left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}}\right) \]

   6. clear-num 53.5%

      \[\leadsto 0.5 \cdot \left(cosTheta_i \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta_O}}}\right) \]

   7. un-div-inv 53.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

8. Applied egg-rr 53.5%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]
9. Step-by-step derivation

   1. associate-*r/ 53.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot cosTheta_i}{\frac{v}{cosTheta_O}}} \]

10. Applied egg-rr 53.5%

    \[\leadsto \color{blue}{\frac{0.5 \cdot cosTheta_i}{\frac{v}{cosTheta_O}}} \]
11. Step-by-step derivation

    1. associate-/l* 54.5%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}} \]

    2. associate-/r* 54.5%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{v}{cosTheta_O \cdot cosTheta_i}}} \]

12. Simplified 54.5%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta_O \cdot cosTheta_i}}} \]

13. Final simplification 54.5%

    \[\leadsto \frac{0.5}{\frac{v}{cosTheta_O \cdot cosTheta_i}} \]

Reproduce

herbie shell --seed 2023318 
(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)))