HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 16.2s

Alternatives: 14

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

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp (/ (* sinTheta_i (- sinTheta_O)) 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_i * -sinTheta_O) / 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_i * -sintheta_o) / 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_i * Float32(-sinTheta_O)) / 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_i * -sinTheta_O) / 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%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (exp (* (- sinTheta_O) (/ sinTheta_i v))) v)
  (* (/ (/ cosTheta_O v) (sinh (/ 1.0 v))) (/ cosTheta_i 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))) / v) * (((cosTheta_O / v) / sinhf((1.0f / v))) * (cosTheta_i / 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))) / v) * (((costheta_o / v) / sinh((1.0e0 / v))) * (costheta_i / 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))) / v) * Float32(Float32(Float32(cosTheta_O / v) / sinh(Float32(Float32(1.0) / v))) * Float32(cosTheta_i / Float32(2.0))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp((-sinTheta_O * (sinTheta_i / v))) / v) * (((cosTheta_O / v) / sinh((single(1.0) / v))) * (cosTheta_i / 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. *-commutative 98.4%

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

   2. times-frac 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. distribute-rgt-neg-out 98.3%

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

   5. associate-*l/ 98.3%

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

   6. *-commutative 98.3%

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

   7. associate-*l/ 98.3%

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

3. Simplified 98.3%

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

   1. expm1-log1p-u 98.3%

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

   2. expm1-udef 49.9%

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

   3. times-frac 49.9%

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

5. Applied egg-rr 49.9%

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

   1. expm1-def 98.2%

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

   2. expm1-log1p 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (exp (* (- sinTheta_O) (/ sinTheta_i v))) 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_O * (sinTheta_i / v))) / 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_o * (sintheta_i / v))) / 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_O) * Float32(sinTheta_i / v))) / v) * Float32(Float32(cosTheta_i * Float32(cosTheta_O / 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))) / v) * ((cosTheta_i * (cosTheta_O / 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. *-commutative 98.4%

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

   2. times-frac 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. distribute-rgt-neg-out 98.3%

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

   5. associate-*l/ 98.3%

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

   6. *-commutative 98.3%

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

   7. associate-*l/ 98.3%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

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))
  (/ (/ cosTheta_i (/ v cosTheta_O)) 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)) * ((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 = (exp((sintheta_i / (v / -sintheta_o))) / (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(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / 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 = (exp((sinTheta_i / (v / -sinTheta_O))) / (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 98.3%

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

   2. exp-neg 98.3%

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

   3. *-commutative 98.3%

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

   4. exp-neg 98.3%

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

   5. distribute-neg-frac 98.3%

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

   6. *-commutative 98.3%

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

   7. distribute-rgt-neg-out 98.3%

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

   8. associate-/l* 98.3%

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

   9. associate-/l* 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (/ (* sinTheta_i (- sinTheta_O)) v)) (* cosTheta_i (/ cosTheta_O 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_i * (cosTheta_O / 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_i * (costheta_o / 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_i * Float32(cosTheta_O / 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_i * (cosTheta_O / 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. *-commutative 98.4%

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

   2. associate-*l/ 98.5%

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

3. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o * ((0.5e0 / (v ** 2.0e0)) * (costheta_i / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((single(0.5) / (v ^ single(2.0))) * (cosTheta_i / sinh((single(1.0) / 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. associate-*r/ 98.4%

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

   2. associate-/l/ 98.4%

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

   3. *-commutative 98.4%

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

   4. /-rgt-identity 98.4%

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

   5. associate-/r/ 98.4%

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

   6. exp-neg 98.4%

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

   7. remove-double-div 98.4%

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

   8. *-commutative 98.4%

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

   9. associate-*l/ 98.4%

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

   10. exp-prod 98.4%

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

   11. associate-*l* 98.4%

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

   12. associate-*l* 98.4%

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

   13. *-commutative 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in sinTheta_O around 0 98.1%

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

   1. expm1-log1p-u 98.1%

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

   2. expm1-udef 50.0%

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

   3. pow-base-1 50.0%

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

   4. add-log-exp 50.0%

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

   5. div-inv 50.0%

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

   6. metadata-eval 50.0%

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

   7. exp-prod 50.0%

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

   8. pow1 50.0%

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

   9. add-log-exp 50.0%

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

   10. *-commutative 50.0%

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

   11. *-commutative 50.0%

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

6. Applied egg-rr 50.0%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p 98.1%

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

   3. associate-/r* 97.3%

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

   4. associate-/l* 97.4%

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

   5. associate-*r* 97.4%

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

   6. unpow2 97.4%

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

8. Simplified 97.4%

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

   1. expm1-log1p-u 97.4%

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

   2. expm1-udef 50.0%

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

   3. associate-/l/ 50.0%

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

   4. *-commutative 50.0%

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

10. Applied egg-rr 50.0%

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

    1. expm1-def 98.2%

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

    2. expm1-log1p 98.2%

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

12. Simplified 98.2%

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

    1. *-commutative 98.2%

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

    2. associate-/l/ 97.4%

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

    3. rem-cube-cbrt 97.0%

       \[\leadsto \frac{\frac{cosTheta_O}{\color{blue}{{\left(\sqrt[3]{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_i}}\right)}^{3}}}}{{v}^{2} \cdot 2} \]

    4. expm1-log1p-u 97.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{cosTheta_O}{{\left(\sqrt[3]{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_i}}\right)}^{3}}}{{v}^{2} \cdot 2}\right)\right)} \]

    5. expm1-udef 50.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{cosTheta_O}{{\left(\sqrt[3]{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_i}}\right)}^{3}}}{{v}^{2} \cdot 2}\right)} - 1} \]

    6. rem-cube-cbrt 50.0%

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

    7. associate-/l/ 50.0%

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

    8. *-commutative 50.0%

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

    9. *-commutative 50.0%

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

14. Applied egg-rr 50.0%

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

    1. expm1-def 98.2%

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

    2. expm1-log1p 98.2%

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

    3. *-rgt-identity 98.2%

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

    4. associate-*r/ 98.2%

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

    5. associate-*l/ 98.2%

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

    6. *-commutative 98.2%

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

    7. associate-/l* 98.1%

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

    8. associate-/r/ 98.2%

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

    9. associate-/r* 98.2%

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

    10. metadata-eval 98.2%

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

16. Simplified 98.2%

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

17. Final simplification 98.2%

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

Alternative 7: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * cosTheta_i) / (sinh((single(1.0) / v)) * (v * (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. associate-*r/ 98.4%

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

   2. associate-/l/ 98.4%

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

   3. *-commutative 98.4%

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

   4. /-rgt-identity 98.4%

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

   5. associate-/r/ 98.4%

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

   6. exp-neg 98.4%

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

   7. remove-double-div 98.4%

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

   8. *-commutative 98.4%

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

   9. associate-*l/ 98.4%

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

   10. exp-prod 98.4%

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

   11. associate-*l* 98.4%

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

   12. associate-*l* 98.4%

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

   13. *-commutative 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in sinTheta_O around 0 98.1%

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

   1. *-un-lft-identity 98.1%

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

   2. pow-base-1 98.1%

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

   3. add-log-exp 50.2%

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

   4. div-inv 50.2%

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

   5. metadata-eval 50.2%

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

   6. exp-prod 50.2%

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

   7. pow1 50.2%

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

   8. add-log-exp 98.1%

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

   9. *-commutative 98.1%

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

   10. *-commutative 98.1%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 8: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O}{\mathsf{fma}\left(2, \frac{v}{cosTheta_i}, \frac{0.3333333333333333}{v \cdot cosTheta_i}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  cosTheta_O
  (fma 2.0 (/ v cosTheta_i) (/ 0.3333333333333333 (* v cosTheta_i)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O / fmaf(2.0f, (v / cosTheta_i), (0.3333333333333333f / (v * cosTheta_i)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O / fma(Float32(2.0), Float32(v / cosTheta_i), Float32(Float32(0.3333333333333333) / Float32(v * 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. associate-*r/ 98.4%

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

   2. associate-/l/ 98.4%

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

   3. *-commutative 98.4%

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

   4. /-rgt-identity 98.4%

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

   5. associate-/r/ 98.4%

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

   6. exp-neg 98.4%

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

   7. remove-double-div 98.4%

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

   8. *-commutative 98.4%

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

   9. associate-*l/ 98.4%

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

   10. exp-prod 98.4%

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

   11. associate-*l* 98.4%

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

   12. associate-*l* 98.4%

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

   13. *-commutative 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in sinTheta_O around 0 98.1%

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

   1. expm1-log1p-u 98.1%

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

   2. expm1-udef 50.0%

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

   3. pow-base-1 50.0%

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

   4. add-log-exp 50.0%

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

   5. div-inv 50.0%

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

   6. metadata-eval 50.0%

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

   7. exp-prod 50.0%

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

   8. pow1 50.0%

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

   9. add-log-exp 50.0%

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

   10. *-commutative 50.0%

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

   11. *-commutative 50.0%

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

6. Applied egg-rr 50.0%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p 98.1%

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

   3. associate-/r* 97.3%

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

   4. associate-/l* 97.4%

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

   5. associate-*r* 97.4%

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

   6. unpow2 97.4%

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

8. Simplified 97.4%

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

   1. expm1-log1p-u 97.4%

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

   2. expm1-udef 50.0%

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

   3. associate-/l/ 50.0%

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

   4. *-commutative 50.0%

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

10. Applied egg-rr 50.0%

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

    1. expm1-def 98.2%

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

    2. expm1-log1p 98.2%

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

12. Simplified 98.2%

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

13. Taylor expanded in v around inf 60.4%

    \[\leadsto \frac{cosTheta_O}{\color{blue}{2 \cdot \frac{v}{cosTheta_i} + 0.3333333333333333 \cdot \frac{1}{cosTheta_i \cdot v}}} \]
14. Step-by-step derivation

    1. fma-def 60.4%

       \[\leadsto \frac{cosTheta_O}{\color{blue}{\mathsf{fma}\left(2, \frac{v}{cosTheta_i}, 0.3333333333333333 \cdot \frac{1}{cosTheta_i \cdot v}\right)}} \]

    2. associate-*r/ 60.4%

       \[\leadsto \frac{cosTheta_O}{\mathsf{fma}\left(2, \frac{v}{cosTheta_i}, \color{blue}{\frac{0.3333333333333333 \cdot 1}{cosTheta_i \cdot v}}\right)} \]

    3. metadata-eval 60.4%

       \[\leadsto \frac{cosTheta_O}{\mathsf{fma}\left(2, \frac{v}{cosTheta_i}, \frac{\color{blue}{0.3333333333333333}}{cosTheta_i \cdot v}\right)} \]

15. Simplified 60.4%

    \[\leadsto \frac{cosTheta_O}{\color{blue}{\mathsf{fma}\left(2, \frac{v}{cosTheta_i}, \frac{0.3333333333333333}{cosTheta_i \cdot v}\right)}} \]

16. Final simplification 60.4%

    \[\leadsto \frac{cosTheta_O}{\mathsf{fma}\left(2, \frac{v}{cosTheta_i}, \frac{0.3333333333333333}{v \cdot cosTheta_i}\right)} \]

Alternative 9: 64.2% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_O}{2 \cdot \frac{v}{cosTheta_i} + 0.3333333333333333 \cdot \frac{1}{v \cdot cosTheta_i}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  cosTheta_O
  (+
   (* 2.0 (/ v cosTheta_i))
   (* 0.3333333333333333 (/ 1.0 (* v cosTheta_i))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O / ((2.0f * (v / cosTheta_i)) + (0.3333333333333333f * (1.0f / (v * 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 = costheta_o / ((2.0e0 * (v / costheta_i)) + (0.3333333333333333e0 * (1.0e0 / (v * costheta_i))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O / Float32(Float32(Float32(2.0) * Float32(v / cosTheta_i)) + Float32(Float32(0.3333333333333333) * Float32(Float32(1.0) / Float32(v * cosTheta_i)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O / ((single(2.0) * (v / cosTheta_i)) + (single(0.3333333333333333) * (single(1.0) / (v * 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. associate-*r/ 98.4%

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

   2. associate-/l/ 98.4%

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

   3. *-commutative 98.4%

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

   4. /-rgt-identity 98.4%

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

   5. associate-/r/ 98.4%

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

   6. exp-neg 98.4%

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

   7. remove-double-div 98.4%

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

   8. *-commutative 98.4%

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

   9. associate-*l/ 98.4%

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

   10. exp-prod 98.4%

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

   11. associate-*l* 98.4%

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

   12. associate-*l* 98.4%

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

   13. *-commutative 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in sinTheta_O around 0 98.1%

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

   1. expm1-log1p-u 98.1%

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

   2. expm1-udef 50.0%

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

   3. pow-base-1 50.0%

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

   4. add-log-exp 50.0%

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

   5. div-inv 50.0%

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

   6. metadata-eval 50.0%

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

   7. exp-prod 50.0%

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

   8. pow1 50.0%

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

   9. add-log-exp 50.0%

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

   10. *-commutative 50.0%

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

   11. *-commutative 50.0%

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

6. Applied egg-rr 50.0%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p 98.1%

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

   3. associate-/r* 97.3%

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

   4. associate-/l* 97.4%

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

   5. associate-*r* 97.4%

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

   6. unpow2 97.4%

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

8. Simplified 97.4%

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

   1. expm1-log1p-u 97.4%

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

   2. expm1-udef 50.0%

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

   3. associate-/l/ 50.0%

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

   4. *-commutative 50.0%

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

10. Applied egg-rr 50.0%

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

    1. expm1-def 98.2%

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

    2. expm1-log1p 98.2%

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

12. Simplified 98.2%

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

13. Taylor expanded in v around inf 60.4%

    \[\leadsto \frac{cosTheta_O}{\color{blue}{2 \cdot \frac{v}{cosTheta_i} + 0.3333333333333333 \cdot \frac{1}{cosTheta_i \cdot v}}} \]

14. Final simplification 60.4%

    \[\leadsto \frac{cosTheta_O}{2 \cdot \frac{v}{cosTheta_i} + 0.3333333333333333 \cdot \frac{1}{v \cdot cosTheta_i}} \]

Alternative 10: 58.5% 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%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 54.8%

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

   1. associate-*r/ 54.8%

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

6. Simplified 54.8%

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

7. Final simplification 54.8%

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

Alternative 11: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 54.8%

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

   1. associate-/l* 54.8%

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

6. Simplified 54.8%

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

7. Final simplification 54.8%

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

Alternative 12: 58.5% 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 98.4%

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

4. Taylor expanded in v around inf 54.8%

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

5. Final simplification 54.8%

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

Alternative 13: 58.5% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta_i}{\frac{v \cdot 2}{cosTheta_O}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i / ((v * single(2.0)) / cosTheta_O);
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 98.4%

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

   1. expm1-log1p-u 98.4%

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

   2. expm1-udef 50.0%

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

   3. associate-/l/ 50.0%

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

   4. associate-*r* 50.0%

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

   5. pow-exp 50.0%

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

   6. associate-*r/ 50.0%

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

   7. pow2 50.0%

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

5. Applied egg-rr 50.0%

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

   1. expm1-def 98.2%

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

   2. expm1-log1p 98.2%

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

   3. *-commutative 98.2%

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

   4. associate-*l* 98.2%

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

   5. associate-*l/ 98.2%

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

   6. associate-*r/ 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in v around inf 54.8%

   \[\leadsto \frac{cosTheta_i}{\color{blue}{2 \cdot \frac{v}{cosTheta_O}}} \]
9. Step-by-step derivation

   1. associate-*r/ 54.8%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{2 \cdot v}{cosTheta_O}}} \]

   2. *-commutative 54.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot 2}}{cosTheta_O}} \]

10. Simplified 54.8%

    \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot 2}{cosTheta_O}}} \]

11. Final simplification 54.8%

    \[\leadsto \frac{cosTheta_i}{\frac{v \cdot 2}{cosTheta_O}} \]

Alternative 14: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O * cosTheta_i) * single(0.5)) / 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 98.4%

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

4. Taylor expanded in v around inf 54.8%

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

   1. associate-*r/ 54.8%

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

   2. *-commutative 54.8%

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

6. Applied egg-rr 54.8%

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

7. Final simplification 54.8%

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

Reproduce

herbie shell --seed 2023311 
(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)))