Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%
Time: 19.1s
Alternatives: 6
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t_1 - \left(eh \cdot \sin t\right) \cdot \sin t_1\right| \end{array} \end{array} \]
(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))
double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

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

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

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

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

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

\begin{array}{l}

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

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 6 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.8% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t_1 - \left(eh \cdot \sin t\right) \cdot \sin t_1\right| \end{array} \end{array} \]
(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))
double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

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

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

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

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

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

\begin{array}{l}

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

Alternative 1: 99.8% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\\ \left|\cos t \cdot \left(ew \cdot \cos t_1\right) - \sin t \cdot \left(eh \cdot \sin t_1\right)\right| \end{array} \end{array} \]
(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (tan t) (/ ew (- eh))))))
   (fabs (- (* (cos t) (* ew (cos t_1))) (* (sin t) (* eh (sin t_1)))))))
double code(double eh, double ew, double t) {
	double t_1 = atan((tan(t) / (ew / -eh)));
	return fabs(((cos(t) * (ew * cos(t_1))) - (sin(t) * (eh * sin(t_1)))));
}

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((tan(t) / (ew / -eh)))
    code = abs(((cos(t) * (ew * cos(t_1))) - (sin(t) * (eh * sin(t_1)))))
end function

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

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

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

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

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

\begin{array}{l}

\\
\begin{array}{l}
t_1 := \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\\
\left|\cos t \cdot \left(ew \cdot \cos t_1\right) - \sin t \cdot \left(eh \cdot \sin t_1\right)\right|
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 99.2% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right) - \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{t}}\right)\right)\right| \end{array} \]
(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (cos t) (* ew (cos (atan (/ (tan t) (/ ew (- eh)))))))
   (* (sin t) (* eh (sin (atan (/ (- eh) (/ ew t)))))))))
double code(double eh, double ew, double t) {
	return fabs(((cos(t) * (ew * cos(atan((tan(t) / (ew / -eh)))))) - (sin(t) * (eh * sin(atan((-eh / (ew / t))))))));
}

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

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

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

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

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

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * N[(ew * N[Cos[N[ArcTan[N[(N[Tan[t], $MachinePrecision] / N[(ew / (-eh)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[((-eh) / N[(ew / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\left|\cos t \cdot \left(ew \cdot \cos \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right) - \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{t}}\right)\right)\right|
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 98.5% accurate, 1.5× speedup?

\[\begin{array}{l} \\ \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right) - \cos t \cdot ew\right| \end{array} \]
(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (* (sin t) (* eh (sin (atan (/ (tan t) (/ ew (- eh)))))))
   (* (cos t) ew))))
double code(double eh, double ew, double t) {
	return fabs(((sin(t) * (eh * sin(atan((tan(t) / (ew / -eh)))))) - (cos(t) * ew)));
}

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

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

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

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

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

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] / N[(ew / (-eh)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\tan t}{\frac{ew}{-eh}}\right)\right) - \cos t \cdot ew\right|
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 98.6% accurate, 1.8× speedup?

\[\begin{array}{l} \\ \left|\cos t \cdot \frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - eh \cdot \sin t\right| \end{array} \]
(FPCore (eh ew t)
 :precision binary64
 (fabs
  (- (* (cos t) (/ ew (hypot 1.0 (* (tan t) (/ eh ew))))) (* eh (sin t)))))
double code(double eh, double ew, double t) {
	return fabs(((cos(t) * (ew / hypot(1.0, (tan(t) * (eh / ew))))) - (eh * sin(t))));
}

public static double code(double eh, double ew, double t) {
	return Math.abs(((Math.cos(t) * (ew / Math.hypot(1.0, (Math.tan(t) * (eh / ew))))) - (eh * Math.sin(t))));
}

def code(eh, ew, t):
	return math.fabs(((math.cos(t) * (ew / math.hypot(1.0, (math.tan(t) * (eh / ew))))) - (eh * math.sin(t))))

function code(eh, ew, t)
	return abs(Float64(Float64(cos(t) * Float64(ew / hypot(1.0, Float64(tan(t) * Float64(eh / ew))))) - Float64(eh * sin(t))))
end

function tmp = code(eh, ew, t)
	tmp = abs(((cos(t) * (ew / hypot(1.0, (tan(t) * (eh / ew))))) - (eh * sin(t))));
end

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * N[(ew / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\left|\cos t \cdot \frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - eh \cdot \sin t\right|
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 98.3% accurate, 1.8× speedup?

\[\begin{array}{l} \\ \left|\cos t \cdot ew - \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{t}}\right)\right)\right| \end{array} \]
(FPCore (eh ew t)
 :precision binary64
 (fabs (- (* (cos t) ew) (* (sin t) (* eh (sin (atan (/ (- eh) (/ ew t)))))))))
double code(double eh, double ew, double t) {
	return fabs(((cos(t) * ew) - (sin(t) * (eh * sin(atan((-eh / (ew / t))))))));
}

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

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

def code(eh, ew, t):
	return math.fabs(((math.cos(t) * ew) - (math.sin(t) * (eh * math.sin(math.atan((-eh / (ew / t))))))))

function code(eh, ew, t)
	return abs(Float64(Float64(cos(t) * ew) - Float64(sin(t) * Float64(eh * sin(atan(Float64(Float64(-eh) / Float64(ew / t))))))))
end

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

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[((-eh) / N[(ew / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\left|\cos t \cdot ew - \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{-eh}{\frac{ew}{t}}\right)\right)\right|
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 98.1% accurate, 2.3× speedup?

\[\begin{array}{l} \\ \left|eh \cdot \sin t + ew \cdot \left(\cos t \cdot \sqrt[3]{-1}\right)\right| \end{array} \]
(FPCore (eh ew t)
 :precision binary64
 (fabs (+ (* eh (sin t)) (* ew (* (cos t) (cbrt -1.0))))))
double code(double eh, double ew, double t) {
	return fabs(((eh * sin(t)) + (ew * (cos(t) * cbrt(-1.0)))));
}

public static double code(double eh, double ew, double t) {
	return Math.abs(((eh * Math.sin(t)) + (ew * (Math.cos(t) * Math.cbrt(-1.0)))));
}

function code(eh, ew, t)
	return abs(Float64(Float64(eh * sin(t)) + Float64(ew * Float64(cos(t) * cbrt(-1.0)))))
end

code[eh_, ew_, t_] := N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\left|eh \cdot \sin t + ew \cdot \left(\cos t \cdot \sqrt[3]{-1}\right)\right|
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2024006 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))