Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 24.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (/ eh ew) (tan t)))))
   (fabs (+ (* (* ew (sin t)) (cos t_1)) (* (* eh (cos t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh / ew) / Math.tan(t)));
	return Math.abs((((ew * Math.sin(t)) * Math.cos(t_1)) + ((eh * Math.cos(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh / ew) / math.tan(t)))
	return math.fabs((((ew * math.sin(t)) * math.cos(t_1)) + ((eh * math.cos(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(t_1)) + Float64(Float64(eh * cos(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh / ew) / tan(t)));
	tmp = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (/ eh ew) (tan t)))))
   (fabs (+ (* (* ew (sin t)) (cos t_1)) (* (* eh (cos t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh / ew) / Math.tan(t)));
	return Math.abs((((ew * Math.sin(t)) * Math.cos(t_1)) + ((eh * Math.cos(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh / ew) / math.tan(t)))
	return math.fabs((((ew * math.sin(t)) * math.cos(t_1)) + ((eh * math.cos(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(t_1)) + Float64(Float64(eh * cos(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh / ew) / tan(t)));
	tmp = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (/ eh ew) (tan t)))))
   (fabs (+ (* (* ew (sin t)) (cos t_1)) (* (* eh (cos t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh / ew) / Math.tan(t)));
	return Math.abs((((ew * Math.sin(t)) * Math.cos(t_1)) + ((eh * Math.cos(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh / ew) / math.tan(t)))
	return math.fabs((((ew * math.sin(t)) * math.cos(t_1)) + ((eh * math.cos(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(t_1)) + Float64(Float64(eh * cos(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh / ew) / tan(t)));
	tmp = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t))))
   (fabs
    (fma
     (/ (sin t) (sqrt (fma t_1 t_1 1.0)))
     ew
     (* eh (* (cos t) (sin (atan (/ eh (* ew (tan t)))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	return fabs(fma((sin(t) / sqrt(fma(t_1, t_1, 1.0))), ew, (eh * (cos(t) * sin(atan((eh / (ew * tan(t)))))))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	return abs(fma(Float64(sin(t) / sqrt(fma(t_1, t_1, 1.0))), ew, Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t)))))))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Sin[t], $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   3. lower-*.f64 99.1

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

5. Applied rewrites 99.1%

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{t \cdot ew}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

   2. lift-*.f64 N/A

      \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   3. lift-*.f64 N/A

      \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right)} \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   4. associate-*l* N/A

      \[\leadsto \left|\color{blue}{ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   5. *-commutative N/A

      \[\leadsto \left|\color{blue}{\left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right) \cdot ew} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   6. lower-fma.f64 N/A

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right), ew, \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}\right| \]

7. Applied rewrites 99.1%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right| \]
8. Add Preprocessing

Alternative 3: 91.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ \mathbf{if}\;t \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} t\_1\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}, \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t))))
   (if (<= t 8.2e+99)
     (fabs
      (fma
       (/ (sin t) (sqrt (fma t_1 t_1 1.0)))
       ew
       (* eh (* (cos t) (sin (atan t_1))))))
     (fabs
      (*
       ew
       (fma
        eh
        (/ (* (cos t) (sin (atan (/ eh (* ew (tan t)))))) ew)
        (sin t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double tmp;
	if (t <= 8.2e+99) {
		tmp = fabs(fma((sin(t) / sqrt(fma(t_1, t_1, 1.0))), ew, (eh * (cos(t) * sin(atan(t_1))))));
	} else {
		tmp = fabs((ew * fma(eh, ((cos(t) * sin(atan((eh / (ew * tan(t)))))) / ew), sin(t))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	tmp = 0.0
	if (t <= 8.2e+99)
		tmp = abs(fma(Float64(sin(t) / sqrt(fma(t_1, t_1, 1.0))), ew, Float64(eh * Float64(cos(t) * sin(atan(t_1))))));
	else
		tmp = abs(Float64(ew * fma(eh, Float64(Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t)))))) / ew), sin(t))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 8.2e+99], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(eh * N[(N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 8.19999999999999959e99

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      3. lower-*.f64 99.2

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   5. Applied rewrites 99.2%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{t \cdot ew}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right)} \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      4. associate-*l* N/A

         \[\leadsto \left|\color{blue}{ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      5. *-commutative N/A

         \[\leadsto \left|\color{blue}{\left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right) \cdot ew} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      6. lower-fma.f64 N/A

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right), ew, \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}\right| \]

   7. Applied rewrites 99.2%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)}\right)\right)\right| \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)}\right)\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right)\right)\right)\right| \]

      3. lower-*.f64 96.3

         \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right)\right)\right)\right| \]

   10. Applied rewrites 96.3%

       \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{t \cdot ew}\right)}\right)\right)\right| \]

   if 8.19999999999999959e99 < t

   1. Initial program 99.6%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      2. +-commutative N/A

         \[\leadsto \left|\color{blue}{\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)} + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(eh \cdot \cos t\right)} \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      5. associate-*l* N/A

         \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)} + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      6. *-commutative N/A

         \[\leadsto \left|\color{blue}{\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right) \cdot eh} + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      7. lower-fma.f64 N/A

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right), eh, \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}\right| \]

   4. Applied rewrites 88.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right), eh, \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \left|\color{blue}{ew \cdot \left(\sin t + \frac{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}{ew}\right)}\right| \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{ew \cdot \left(\sin t + \frac{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}{ew}\right)}\right| \]

      2. +-commutative N/A

         \[\leadsto \left|ew \cdot \color{blue}{\left(\frac{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)}{ew} + \sin t\right)}\right| \]

      3. associate-/l* N/A

         \[\leadsto \left|ew \cdot \left(\color{blue}{eh \cdot \frac{\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}} + \sin t\right)\right| \]

      4. lower-fma.f64 N/A

         \[\leadsto \left|ew \cdot \color{blue}{\mathsf{fma}\left(eh, \frac{\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}, \sin t\right)}\right| \]

   7. Applied rewrites 88.4%

      \[\leadsto \left|\color{blue}{ew \cdot \mathsf{fma}\left(eh, \frac{\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}, \sin t\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}, \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} t\_1\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t))))
   (fabs
    (fma
     (/ (sin t) (sqrt (fma t_1 t_1 1.0)))
     ew
     (* eh (* (cos t) (sin (atan t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	return fabs(fma((sin(t) / sqrt(fma(t_1, t_1, 1.0))), ew, (eh * (cos(t) * sin(atan(t_1))))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	return abs(fma(Float64(sin(t) / sqrt(fma(t_1, t_1, 1.0))), ew, Float64(eh * Float64(cos(t) * sin(atan(t_1))))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Sin[t], $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   3. lower-*.f64 99.1

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

5. Applied rewrites 99.1%

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{t \cdot ew}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right| \]

   2. lift-*.f64 N/A

      \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   3. lift-*.f64 N/A

      \[\leadsto \left|\color{blue}{\left(ew \cdot \sin t\right)} \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   4. associate-*l* N/A

      \[\leadsto \left|\color{blue}{ew \cdot \left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   5. *-commutative N/A

      \[\leadsto \left|\color{blue}{\left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right) \cdot ew} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   6. lower-fma.f64 N/A

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\sin t \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right), ew, \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right)}\right| \]

7. Applied rewrites 99.1%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}\right| \]

8. Taylor expanded in t around 0

   \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)}\right)\right)\right| \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)}\right)\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right)\right)\right)\right| \]

   3. lower-*.f64 91.0

      \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{\color{blue}{t \cdot ew}}\right)\right)\right)\right| \]

10. Applied rewrites 91.0%

    \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{t \cdot ew}\right)}\right)\right)\right| \]

11. Final simplification 91.0%

    \[\leadsto \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, \frac{eh}{ew \cdot t}, 1\right)}}, ew, eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right)\right| \]
12. Add Preprocessing

Alternative 5: 73.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -2.2 \cdot 10^{+140}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{0.5}{t\_1}, eh \cdot eh, t\_1\right)\right|\\ \mathbf{elif}\;ew \leq 1.25 \cdot 10^{+97}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= ew -2.2e+140)
     (fabs (fma (/ 0.5 t_1) (* eh eh) t_1))
     (if (<= ew 1.25e+97) (fabs (* eh (cos t))) (fabs t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if (ew <= -2.2e+140) {
		tmp = fabs(fma((0.5 / t_1), (eh * eh), t_1));
	} else if (ew <= 1.25e+97) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -2.2e+140)
		tmp = abs(fma(Float64(0.5 / t_1), Float64(eh * eh), t_1));
	elseif (ew <= 1.25e+97)
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(t_1);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -2.2e+140], N[Abs[N[(N[(0.5 / t$95$1), $MachinePrecision] * N[(eh * eh), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.25e+97], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.1999999999999998e140

   1. Initial program 99.7%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 79.4%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{eh}{ew \cdot \tan t}, \frac{1}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \left(eh \cdot \cos t\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t + {eh}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t} + \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)}\right| \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, {eh}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t} + \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

      2. lower-sin.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\sin t}, {eh}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t} + \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \color{blue}{{eh}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t} + \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)}\right)\right| \]

      4. unpow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \color{blue}{\left(eh \cdot eh\right)} \cdot \left(\frac{-1}{2} \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t} + \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \color{blue}{\left(eh \cdot eh\right)} \cdot \left(\frac{-1}{2} \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t} + \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)\right| \]

      6. distribute-lft1-in N/A

         \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \color{blue}{\left(\left(\frac{-1}{2} + 1\right) \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)}\right)\right| \]

      7. metadata-eval N/A

         \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)}\right)\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{{\cos t}^{2}}{ew \cdot \sin t}}\right)\right)\right| \]

      10. lower-pow.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(\frac{1}{2} \cdot \frac{\color{blue}{{\cos t}^{2}}}{ew \cdot \sin t}\right)\right)\right| \]

      11. lower-cos.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(\frac{1}{2} \cdot \frac{{\color{blue}{\cos t}}^{2}}{ew \cdot \sin t}\right)\right)\right| \]

      12. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{\color{blue}{ew \cdot \sin t}}\right)\right)\right| \]

      13. lower-sin.f64 75.9

          \[\leadsto \left|\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \color{blue}{\sin t}}\right)\right)\right| \]

   7. Applied rewrites 75.9%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \left(0.5 \cdot \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 75.9%

      \[\leadsto \left|\mathsf{fma}\left(\frac{\mathsf{fma}\left(\cos \left(t + t\right), 0.25, 0.25\right)}{\sin t \cdot ew}, \color{blue}{eh \cdot eh}, \sin t \cdot ew\right)\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\mathsf{fma}\left(\frac{\frac{1}{2}}{\sin t \cdot ew}, eh \cdot eh, \sin t \cdot ew\right)\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 75.9%

       \[\leadsto \left|\mathsf{fma}\left(\frac{0.5}{\sin t \cdot ew}, eh \cdot eh, \sin t \cdot ew\right)\right| \]

   if -2.1999999999999998e140 < ew < 1.25e97

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 30.4%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{eh}{ew \cdot \tan t}, \frac{1}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \left(eh \cdot \cos t\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}\right| \]

   5. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]

      2. lower-cos.f64 80.6

         \[\leadsto \left|eh \cdot \color{blue}{\cos t}\right| \]

   7. Applied rewrites 80.6%

      \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]

   if 1.25e97 < ew

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 80.2%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{eh}{ew \cdot \tan t}, \frac{1}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \left(eh \cdot \cos t\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]

      2. lower-sin.f64 78.7

         \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

   7. Applied rewrites 78.7%

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.2 \cdot 10^{+140}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{0.5}{ew \cdot \sin t}, eh \cdot eh, ew \cdot \sin t\right)\right|\\ \mathbf{elif}\;ew \leq 1.25 \cdot 10^{+97}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -2.2 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.25 \cdot 10^{+97}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= ew -2.2e+140) t_1 (if (<= ew 1.25e+97) (fabs (* eh (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -2.2e+140) {
		tmp = t_1;
	} else if (ew <= 1.25e+97) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * sin(t)))
    if (ew <= (-2.2d+140)) then
        tmp = t_1
    else if (ew <= 1.25d+97) then
        tmp = abs((eh * cos(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.sin(t)));
	double tmp;
	if (ew <= -2.2e+140) {
		tmp = t_1;
	} else if (ew <= 1.25e+97) {
		tmp = Math.abs((eh * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -2.2e+140:
		tmp = t_1
	elif ew <= 1.25e+97:
		tmp = math.fabs((eh * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -2.2e+140)
		tmp = t_1;
	elseif (ew <= 1.25e+97)
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -2.2e+140)
		tmp = t_1;
	elseif (ew <= 1.25e+97)
		tmp = abs((eh * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.2e+140], t$95$1, If[LessEqual[ew, 1.25e+97], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.1999999999999998e140 or 1.25e97 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 79.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{eh}{ew \cdot \tan t}, \frac{1}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \left(eh \cdot \cos t\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}\right| \]

   5. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]

      2. lower-sin.f64 77.2

         \[\leadsto \left|ew \cdot \color{blue}{\sin t}\right| \]

   7. Applied rewrites 77.2%

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]

   if -2.1999999999999998e140 < ew < 1.25e97

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Add Preprocessing

   4. Applied rewrites 30.4%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{eh}{ew \cdot \tan t}, \frac{1}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \left(eh \cdot \cos t\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}\right| \]

   5. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]

      2. lower-cos.f64 80.6

         \[\leadsto \left|eh \cdot \color{blue}{\cos t}\right| \]

   7. Applied rewrites 80.6%

      \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot \cos t\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* eh (cos t))))

C

double code(double eh, double ew, double t) {
	return fabs((eh * cos(t)));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((eh * cos(t)))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((eh * Math.cos(t)));
}

Python

def code(eh, ew, t):
	return math.fabs((eh * math.cos(t)))

Julia

function code(eh, ew, t)
	return abs(Float64(eh * cos(t)))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((eh * cos(t)));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. Add Preprocessing

4. Applied rewrites 46.6%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{eh}{ew \cdot \tan t}, \frac{1}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \left(eh \cdot \cos t\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}\right| \]

5. Taylor expanded in eh around inf

   \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]

   2. lower-cos.f64 63.1

      \[\leadsto \left|eh \cdot \color{blue}{\cos t}\right| \]

7. Applied rewrites 63.1%

   \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]
8. Add Preprocessing

Alternative 8: 41.7% accurate, 108.8× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot 1\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* eh 1.0)))

C

double code(double eh, double ew, double t) {
	return fabs((eh * 1.0));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((eh * 1.0d0))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((eh * 1.0));
}

Python

def code(eh, ew, t):
	return math.fabs((eh * 1.0))

Julia

function code(eh, ew, t)
	return abs(Float64(eh * 1.0))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((eh * 1.0));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(eh * 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. Add Preprocessing

4. Applied rewrites 46.6%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{eh}{ew \cdot \tan t}, \frac{1}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \left(eh \cdot \cos t\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}\right| \]

5. Taylor expanded in eh around inf

   \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]

   2. lower-cos.f64 63.1

      \[\leadsto \left|eh \cdot \color{blue}{\cos t}\right| \]

7. Applied rewrites 63.1%

   \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]

8. Taylor expanded in t around 0

   \[\leadsto \left|eh \cdot 1\right| \]
9. Step-by-step derivation

10. Applied rewrites 43.1%

    \[\leadsto \left|eh \cdot 1\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024237 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))