HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 16.6s

Alternatives: 15

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

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i (- sinTheta_O)) v))
   (* (/ 1.0 v) (* cosTheta_i cosTheta_O)))
  (* 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_i * -sinTheta_O) / v)) * ((1.0f / v) * (cosTheta_i * cosTheta_O))) / (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_i * -sintheta_o) / v)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / (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_i * Float32(-sinTheta_O)) / v)) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O))) / 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_i * -sinTheta_O) / v)) * ((single(1.0) / v) * (cosTheta_i * cosTheta_O))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
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. Step-by-step derivation

   1. div-inv 98.9%

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

3. Applied egg-rr 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (exp (/ sinTheta_i (/ v (- sinTheta_O)))) (* (sinh (/ 1.0 v)) 2.0))
  (* (/ 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 (expf((sinTheta_i / (v / -sinTheta_O))) / (sinhf((1.0f / v)) * 2.0f)) * ((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 = (exp((sintheta_i / (v / -sintheta_o))) / (sinh((1.0e0 / v)) * 2.0e0)) * ((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(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))) * 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 = (exp((sinTheta_i / (v / -sinTheta_O))) / (sinh((single(1.0) / v)) * single(2.0))) * ((single(1.0) / v) * (cosTheta_O * (cosTheta_i / 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. Step-by-step derivation

   1. times-frac 98.7%

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

   2. exp-neg 98.7%

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

   3. *-commutative 98.7%

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

   4. exp-neg 98.7%

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

   5. distribute-neg-frac 98.7%

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

   6. *-commutative 98.7%

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

   7. distribute-rgt-neg-out 98.7%

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

   8. associate-/l* 98.7%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.8%

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

   2. associate-/r/ 98.8%

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

5. Applied egg-rr 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* sinTheta_i (- sinTheta_O)) v)) (* cosTheta_O (/ cosTheta_i 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_i * -sinTheta_O) / v)) * (cosTheta_O * (cosTheta_i / 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_i * -sintheta_o) / v)) * (costheta_o * (costheta_i / 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_i * Float32(-sinTheta_O)) / v)) * Float32(cosTheta_O * Float32(cosTheta_i / 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_i * -sinTheta_O) / v)) * (cosTheta_O * (cosTheta_i / v))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
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. Step-by-step derivation

   1. associate-/l* 98.8%

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

   2. associate-/r/ 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. times-frac 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.6%

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

   4. clear-num 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 6: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \frac{{v}^{2}}{cosTheta_O}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_i / (sinhf((1.0f / v)) * (powf(v, 2.0f) / cosTheta_O)));
}

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_i / (sinh((1.0e0 / v)) * ((v ** 2.0e0) / costheta_o)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(cosTheta_i / Float32(sinh(Float32(Float32(1.0) / v)) * Float32((v ^ Float32(2.0)) / cosTheta_O))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (cosTheta_i / (sinh((single(1.0) / v)) * ((v ^ single(2.0)) / cosTheta_O)));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.6%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {1}^{sinTheta_i}\right)}\right)\right)} \]

   2. expm1-udef 53.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {1}^{sinTheta_i}\right)}\right)} - 1} \]

   3. *-un-lft-identity 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {1}^{sinTheta_i}\right)}\right)} - 1 \]

   4. *-commutative 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\color{blue}{\left(2 \cdot {1}^{sinTheta_i}\right) \cdot \sinh \left(\frac{1}{v}\right)}}\right)} - 1 \]

   5. pow-base-1 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\left(2 \cdot \color{blue}{1}\right) \cdot \sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   6. metadata-eval 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\color{blue}{2} \cdot \sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   7. times-frac 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{2} \cdot \frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}}\right)} - 1 \]

   8. metadata-eval 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5} \cdot \frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   9. associate-*r/ 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O}}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   10. pow2 53.3%

       \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\frac{\color{blue}{{v}^{2}}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

6. Applied egg-rr 53.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\frac{{v}^{2}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\frac{{v}^{2}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \]

   2. expm1-log1p 98.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{cosTheta_i}{\frac{{v}^{2}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}} \]

   3. associate-/l/ 98.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \frac{{v}^{2}}{cosTheta_O}}} \]

8. Simplified 98.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \frac{{v}^{2}}{cosTheta_O}}} \]

9. Final simplification 98.5%

   \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \frac{{v}^{2}}{cosTheta_O}} \]

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.6%

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

   1. div-inv 98.5%

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

   2. pow-base-1 98.5%

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

   3. metadata-eval 98.5%

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

   4. associate-*r/ 98.5%

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

   5. pow2 98.5%

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

6. Applied egg-rr 98.5%

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

   1. associate-*r/ 98.6%

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

   2. associate-*l/ 98.6%

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

   3. associate-*r/ 98.5%

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

   4. associate-*l/ 98.5%

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

   5. associate-/r* 98.5%

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

   6. associate-*l/ 98.3%

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

   7. associate-*r/ 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-*l/ 98.6%

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

   10. *-commutative 98.6%

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

   11. *-rgt-identity 98.6%

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

8. Simplified 98.6%

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

9. Final simplification 98.6%

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

Alternative 8: 70.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta_i}{\left(\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{v \cdot cosTheta_O}\right) + \frac{0.008333333333333333}{cosTheta_O \cdot {v}^{3}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  0.5
  (/
   cosTheta_i
   (+
    (+ (/ v cosTheta_O) (/ 0.16666666666666666 (* v cosTheta_O)))
    (/ 0.008333333333333333 (* cosTheta_O (pow v 3.0)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_i / (((v / cosTheta_O) + (0.16666666666666666f / (v * cosTheta_O))) + (0.008333333333333333f / (cosTheta_O * powf(v, 3.0f)))));
}

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 = 0.5e0 * (costheta_i / (((v / costheta_o) + (0.16666666666666666e0 / (v * costheta_o))) + (0.008333333333333333e0 / (costheta_o * (v ** 3.0e0)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(cosTheta_i / Float32(Float32(Float32(v / cosTheta_O) + Float32(Float32(0.16666666666666666) / Float32(v * cosTheta_O))) + Float32(Float32(0.008333333333333333) / Float32(cosTheta_O * (v ^ Float32(3.0)))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (cosTheta_i / (((v / cosTheta_O) + (single(0.16666666666666666) / (v * cosTheta_O))) + (single(0.008333333333333333) / (cosTheta_O * (v ^ single(3.0))))));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.6%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {1}^{sinTheta_i}\right)}\right)\right)} \]

   2. expm1-udef 53.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {1}^{sinTheta_i}\right)}\right)} - 1} \]

   3. *-un-lft-identity 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {1}^{sinTheta_i}\right)}\right)} - 1 \]

   4. *-commutative 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\color{blue}{\left(2 \cdot {1}^{sinTheta_i}\right) \cdot \sinh \left(\frac{1}{v}\right)}}\right)} - 1 \]

   5. pow-base-1 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\left(2 \cdot \color{blue}{1}\right) \cdot \sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   6. metadata-eval 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\color{blue}{2} \cdot \sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   7. times-frac 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{2} \cdot \frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}}\right)} - 1 \]

   8. metadata-eval 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5} \cdot \frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   9. associate-*r/ 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O}}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   10. pow2 53.3%

       \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\frac{\color{blue}{{v}^{2}}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

6. Applied egg-rr 53.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\frac{{v}^{2}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\frac{{v}^{2}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \]

   2. expm1-log1p 98.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{cosTheta_i}{\frac{{v}^{2}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}} \]

   3. associate-/l/ 98.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \frac{{v}^{2}}{cosTheta_O}}} \]

8. Simplified 98.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \frac{{v}^{2}}{cosTheta_O}}} \]

9. Taylor expanded in v around inf 71.8%

   \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\color{blue}{0.008333333333333333 \cdot \frac{1}{cosTheta_O \cdot {v}^{3}} + \left(0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}\right)}} \]
10. Step-by-step derivation

    1. +-commutative 71.8%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\color{blue}{\left(0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}\right) + 0.008333333333333333 \cdot \frac{1}{cosTheta_O \cdot {v}^{3}}}} \]

    2. +-commutative 71.8%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\color{blue}{\left(\frac{v}{cosTheta_O} + 0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v}\right)} + 0.008333333333333333 \cdot \frac{1}{cosTheta_O \cdot {v}^{3}}} \]

    3. associate-*r/ 71.8%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\left(\frac{v}{cosTheta_O} + \color{blue}{\frac{0.16666666666666666 \cdot 1}{cosTheta_O \cdot v}}\right) + 0.008333333333333333 \cdot \frac{1}{cosTheta_O \cdot {v}^{3}}} \]

    4. metadata-eval 71.8%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\left(\frac{v}{cosTheta_O} + \frac{\color{blue}{0.16666666666666666}}{cosTheta_O \cdot v}\right) + 0.008333333333333333 \cdot \frac{1}{cosTheta_O \cdot {v}^{3}}} \]

    5. *-commutative 71.8%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\left(\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{\color{blue}{v \cdot cosTheta_O}}\right) + 0.008333333333333333 \cdot \frac{1}{cosTheta_O \cdot {v}^{3}}} \]

    6. associate-*r/ 71.8%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\left(\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{v \cdot cosTheta_O}\right) + \color{blue}{\frac{0.008333333333333333 \cdot 1}{cosTheta_O \cdot {v}^{3}}}} \]

    7. metadata-eval 71.8%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\left(\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{v \cdot cosTheta_O}\right) + \frac{\color{blue}{0.008333333333333333}}{cosTheta_O \cdot {v}^{3}}} \]

    8. *-commutative 71.8%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\left(\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{v \cdot cosTheta_O}\right) + \frac{0.008333333333333333}{\color{blue}{{v}^{3} \cdot cosTheta_O}}} \]

11. Simplified 71.8%

    \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\color{blue}{\left(\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{v \cdot cosTheta_O}\right) + \frac{0.008333333333333333}{{v}^{3} \cdot cosTheta_O}}} \]

12. Final simplification 71.8%

    \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\left(\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{v \cdot cosTheta_O}\right) + \frac{0.008333333333333333}{cosTheta_O \cdot {v}^{3}}} \]

Alternative 9: 64.3% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{v \cdot cosTheta_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  0.5
  (/
   cosTheta_i
   (+ (/ v cosTheta_O) (/ 0.16666666666666666 (* v cosTheta_O))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_i / ((v / cosTheta_O) + (0.16666666666666666f / (v * cosTheta_O))));
}

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_i / ((v / costheta_o) + (0.16666666666666666e0 / (v * costheta_o))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(cosTheta_i / Float32(Float32(v / cosTheta_O) + Float32(Float32(0.16666666666666666) / Float32(v * cosTheta_O)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (cosTheta_i / ((v / cosTheta_O) + (single(0.16666666666666666) / (v * cosTheta_O))));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in sinTheta_O around 0 98.6%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {1}^{sinTheta_i}\right)}\right)\right)} \]

   2. expm1-udef 53.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {1}^{sinTheta_i}\right)}\right)} - 1} \]

   3. *-un-lft-identity 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot {1}^{sinTheta_i}\right)}\right)} - 1 \]

   4. *-commutative 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\color{blue}{\left(2 \cdot {1}^{sinTheta_i}\right) \cdot \sinh \left(\frac{1}{v}\right)}}\right)} - 1 \]

   5. pow-base-1 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\left(2 \cdot \color{blue}{1}\right) \cdot \sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   6. metadata-eval 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\color{blue}{2} \cdot \sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   7. times-frac 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{2} \cdot \frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}}\right)} - 1 \]

   8. metadata-eval 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5} \cdot \frac{\frac{cosTheta_i}{v \cdot \frac{v}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   9. associate-*r/ 53.3%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O}}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

   10. pow2 53.3%

       \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\frac{\color{blue}{{v}^{2}}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1 \]

6. Applied egg-rr 53.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\frac{{v}^{2}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 98.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{\frac{cosTheta_i}{\frac{{v}^{2}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \]

   2. expm1-log1p 98.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{cosTheta_i}{\frac{{v}^{2}}{cosTheta_O}}}{\sinh \left(\frac{1}{v}\right)}} \]

   3. associate-/l/ 98.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \frac{{v}^{2}}{cosTheta_O}}} \]

8. Simplified 98.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \frac{{v}^{2}}{cosTheta_O}}} \]

9. Taylor expanded in v around inf 65.3%

   \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\color{blue}{0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}}} \]
10. Step-by-step derivation

    1. +-commutative 65.3%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\color{blue}{\frac{v}{cosTheta_O} + 0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v}}} \]

    2. associate-*r/ 65.3%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \color{blue}{\frac{0.16666666666666666 \cdot 1}{cosTheta_O \cdot v}}} \]

    3. metadata-eval 65.3%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \frac{\color{blue}{0.16666666666666666}}{cosTheta_O \cdot v}} \]

    4. *-commutative 65.3%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{\color{blue}{v \cdot cosTheta_O}}} \]

11. Simplified 65.3%

    \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\color{blue}{\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{v \cdot cosTheta_O}}} \]

12. Final simplification 65.3%

    \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \frac{0.16666666666666666}{v \cdot cosTheta_O}} \]

Alternative 10: 59.1% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{v}{cosTheta_O} \cdot \frac{1}{cosTheta_i}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (* (/ v cosTheta_O) (/ 1.0 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) * (1.0f / 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) * (1.0e0 / costheta_i))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / ((v / cosTheta_O) * (single(1.0) / cosTheta_i));
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. Step-by-step derivation

   1. times-frac 98.7%

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

   2. exp-neg 98.7%

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

   3. *-commutative 98.7%

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

   4. exp-neg 98.7%

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

   5. distribute-neg-frac 98.7%

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

   6. *-commutative 98.7%

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

   7. distribute-rgt-neg-out 98.7%

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

   8. associate-/l* 98.7%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.8%

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

   2. associate-/r/ 98.8%

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

5. Applied egg-rr 98.8%

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

6. Taylor expanded in v around inf 59.2%

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

   1. associate-*r/ 59.2%

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

   2. associate-/l* 59.3%

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

   3. associate-/r* 59.3%

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

8. Simplified 59.3%

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

   1. div-inv 59.3%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{v}{cosTheta_O} \cdot \frac{1}{cosTheta_i}}} \]

10. Applied egg-rr 59.3%

    \[\leadsto \frac{0.5}{\color{blue}{\frac{v}{cosTheta_O} \cdot \frac{1}{cosTheta_i}}} \]

11. Final simplification 59.3%

    \[\leadsto \frac{0.5}{\frac{v}{cosTheta_O} \cdot \frac{1}{cosTheta_i}} \]

Alternative 11: 59.1% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{cosTheta_i \cdot \left(cosTheta_O \cdot 0.5\right)}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (v / (cosTheta_i * (cosTheta_O * 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 / (costheta_i * (costheta_o * 0.5e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (v / (cosTheta_i * (cosTheta_O * single(0.5))));
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. Step-by-step derivation

   1. times-frac 98.7%

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

   2. exp-neg 98.7%

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

   3. *-commutative 98.7%

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

   4. exp-neg 98.7%

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

   5. distribute-neg-frac 98.7%

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

   6. *-commutative 98.7%

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

   7. distribute-rgt-neg-out 98.7%

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

   8. associate-/l* 98.7%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.8%

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

   2. associate-/r/ 98.8%

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

5. Applied egg-rr 98.8%

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

6. Taylor expanded in v around inf 59.2%

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

   1. associate-*r/ 59.2%

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

   2. associate-/l* 59.3%

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

   3. associate-/r* 59.3%

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

8. Simplified 59.3%

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

   1. clear-num 59.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}{0.5}}} \]

   2. inv-pow 59.5%

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

   3. associate-/l/ 59.5%

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

10. Applied egg-rr 59.5%

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

    1. unpow-1 59.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta_i \cdot cosTheta_O}}{0.5}}} \]

    2. associate-/l/ 59.5%

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

    3. *-commutative 59.5%

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

    4. associate-*r* 59.5%

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

12. Simplified 59.5%

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

13. Final simplification 59.5%

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

Alternative 12: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * (cosTheta_i / v)) * single(0.5);
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. Step-by-step derivation

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.7%

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

   3. *-commutative 98.7%

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

   4. /-rgt-identity 98.7%

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

   5. associate-/r/ 98.7%

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

   6. exp-neg 98.7%

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

   7. remove-double-div 98.7%

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

   8. *-commutative 98.7%

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

   9. associate-*l/ 98.7%

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

   10. exp-prod 98.7%

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

   11. associate-*l* 98.7%

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

   12. associate-*l* 98.8%

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

   13. *-commutative 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in v around inf 59.2%

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

   1. associate-*r/ 59.2%

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

6. Simplified 59.2%

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

7. Final simplification 59.2%

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

Alternative 13: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((cosTheta_i * cosTheta_O) / 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. Step-by-step derivation

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.7%

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

   3. *-commutative 98.7%

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

   4. /-rgt-identity 98.7%

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

   5. associate-/r/ 98.7%

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

   6. exp-neg 98.7%

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

   7. remove-double-div 98.7%

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

   8. *-commutative 98.7%

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

   9. associate-*l/ 98.7%

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

   10. exp-prod 98.7%

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

   11. associate-*l* 98.7%

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

   12. associate-*l* 98.8%

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

   13. *-commutative 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in v around inf 59.2%

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

5. Final simplification 59.2%

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

Alternative 14: 59.1% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

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_i * costheta_o))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (v / (cosTheta_i * cosTheta_O));
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. Step-by-step derivation

   1. times-frac 98.7%

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

   2. exp-neg 98.7%

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

   3. *-commutative 98.7%

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

   4. exp-neg 98.7%

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

   5. distribute-neg-frac 98.7%

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

   6. *-commutative 98.7%

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

   7. distribute-rgt-neg-out 98.7%

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

   8. associate-/l* 98.7%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.8%

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

   2. associate-/r/ 98.8%

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

5. Applied egg-rr 98.8%

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

6. Taylor expanded in v around inf 59.2%

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

   1. associate-*r/ 59.2%

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

   2. associate-/l* 59.3%

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

   3. associate-/r* 59.3%

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

8. Simplified 59.3%

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

9. Taylor expanded in v around 0 59.3%

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

10. Final simplification 59.3%

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

Alternative 15: 59.1% accurate, 31.4× speedup

Math

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

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(Float32(v / 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 98.7%

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

   1. times-frac 98.7%

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

   2. exp-neg 98.7%

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

   3. *-commutative 98.7%

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

   4. exp-neg 98.7%

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

   5. distribute-neg-frac 98.7%

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

   6. *-commutative 98.7%

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

   7. distribute-rgt-neg-out 98.7%

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

   8. associate-/l* 98.7%

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

   9. associate-/l* 98.6%

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

3. Simplified 98.6%

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

   1. div-inv 98.8%

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

   2. associate-/r/ 98.8%

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

5. Applied egg-rr 98.8%

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

6. Taylor expanded in v around inf 59.2%

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

   1. associate-*r/ 59.2%

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

   2. associate-/l* 59.3%

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

   3. associate-/r* 59.3%

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

8. Simplified 59.3%

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

9. Final simplification 59.3%

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

Reproduce

herbie shell --seed 2023300 
(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)))