tanhf (example 3.4)

Average Error: 29.7 → 0.7

Time: 18.0s

Precision: binary64

Cost: 65672

Math

\[\frac{1 - \cos x}{\sin x} \]

↓

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{\sin x}\\ \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0.0005:\\ \;\;\;\;0.5 \cdot x + \left(0.041666666666666664 \cdot {x}^{3} + 0.004166666666666667 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x + -1}{\sin x \cdot \left(\sin x \cdot \frac{\sin x}{\sin x \cdot \sin \left(-x\right)}\right)}\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (cos x)) (sin x))))
   (if (<= t_0 -0.02)
     t_0
     (if (<= t_0 0.0005)
       (+
        (* 0.5 x)
        (+
         (* 0.041666666666666664 (pow x 3.0))
         (* 0.004166666666666667 (pow x 5.0))))
       (/
        (+ (cos x) -1.0)
        (* (sin x) (* (sin x) (/ (sin x) (* (sin x) (sin (- x)))))))))))

C

double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}

↓

double code(double x) {
	double t_0 = (1.0 - cos(x)) / sin(x);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0;
	} else if (t_0 <= 0.0005) {
		tmp = (0.5 * x) + ((0.041666666666666664 * pow(x, 3.0)) + (0.004166666666666667 * pow(x, 5.0)));
	} else {
		tmp = (cos(x) + -1.0) / (sin(x) * (sin(x) * (sin(x) / (sin(x) * sin(-x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / sin(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - cos(x)) / sin(x)
    if (t_0 <= (-0.02d0)) then
        tmp = t_0
    else if (t_0 <= 0.0005d0) then
        tmp = (0.5d0 * x) + ((0.041666666666666664d0 * (x ** 3.0d0)) + (0.004166666666666667d0 * (x ** 5.0d0)))
    else
        tmp = (cos(x) + (-1.0d0)) / (sin(x) * (sin(x) * (sin(x) / (sin(x) * sin(-x)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / Math.sin(x);
}

↓

public static double code(double x) {
	double t_0 = (1.0 - Math.cos(x)) / Math.sin(x);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0;
	} else if (t_0 <= 0.0005) {
		tmp = (0.5 * x) + ((0.041666666666666664 * Math.pow(x, 3.0)) + (0.004166666666666667 * Math.pow(x, 5.0)));
	} else {
		tmp = (Math.cos(x) + -1.0) / (Math.sin(x) * (Math.sin(x) * (Math.sin(x) / (Math.sin(x) * Math.sin(-x)))));
	}
	return tmp;
}

Python

def code(x):
	return (1.0 - math.cos(x)) / math.sin(x)

↓

def code(x):
	t_0 = (1.0 - math.cos(x)) / math.sin(x)
	tmp = 0
	if t_0 <= -0.02:
		tmp = t_0
	elif t_0 <= 0.0005:
		tmp = (0.5 * x) + ((0.041666666666666664 * math.pow(x, 3.0)) + (0.004166666666666667 * math.pow(x, 5.0)))
	else:
		tmp = (math.cos(x) + -1.0) / (math.sin(x) * (math.sin(x) * (math.sin(x) / (math.sin(x) * math.sin(-x)))))
	return tmp

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / sin(x))
end

↓

function code(x)
	t_0 = Float64(Float64(1.0 - cos(x)) / sin(x))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = t_0;
	elseif (t_0 <= 0.0005)
		tmp = Float64(Float64(0.5 * x) + Float64(Float64(0.041666666666666664 * (x ^ 3.0)) + Float64(0.004166666666666667 * (x ^ 5.0))));
	else
		tmp = Float64(Float64(cos(x) + -1.0) / Float64(sin(x) * Float64(sin(x) * Float64(sin(x) / Float64(sin(x) * sin(Float64(-x)))))));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / sin(x);
end

↓

function tmp_2 = code(x)
	t_0 = (1.0 - cos(x)) / sin(x);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = t_0;
	elseif (t_0 <= 0.0005)
		tmp = (0.5 * x) + ((0.041666666666666664 * (x ^ 3.0)) + (0.004166666666666667 * (x ^ 5.0)));
	else
		tmp = (cos(x) + -1.0) / (sin(x) * (sin(x) * (sin(x) / (sin(x) * sin(-x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], t$95$0, If[LessEqual[t$95$0, 0.0005], N[(N[(0.5 * x), $MachinePrecision] + N[(N[(0.041666666666666664 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.004166666666666667 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * N[Sin[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	29.7
Target	0.0
Herbie	0.7

\[\tan \left(\frac{x}{2}\right) \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x)) < -0.0200000000000000004

   1. Initial program 0.8

      \[\frac{1 - \cos x}{\sin x} \]

   if -0.0200000000000000004 < (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x)) < 5.0000000000000001e-4

   1. Initial program 59.7

      \[\frac{1 - \cos x}{\sin x} \]

   2. Taylor expanded in x around 0 0.4

      \[\leadsto \color{blue}{0.5 \cdot x + \left(0.004166666666666667 \cdot {x}^{5} + 0.041666666666666664 \cdot {x}^{3}\right)} \]

   3. Simplified 0.4

      \[\leadsto \color{blue}{0.5 \cdot x + \left(0.041666666666666664 \cdot {x}^{3} + 0.004166666666666667 \cdot {x}^{5}\right)} \]
      Proof

      [Start] 0.4	\[ 0.5 \cdot x + \left(0.004166666666666667 \cdot {x}^{5} + 0.041666666666666664 \cdot {x}^{3}\right) \]
      rational_best-simplify-1 [=>] 0.4	\[ 0.5 \cdot x + \color{blue}{\left(0.041666666666666664 \cdot {x}^{3} + 0.004166666666666667 \cdot {x}^{5}\right)} \]

   if 5.0000000000000001e-4 < (/.f64 (-.f64 1 (cos.f64 x)) (sin.f64 x))

   1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x} \]

   2. Applied egg-rr 1.3

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot 3 + \frac{1 - \cos x}{\sin x} \cdot -2} \]

   3. Applied egg-rr 0.9

      \[\leadsto \color{blue}{\frac{\frac{\cos x + -1}{\sin x}}{\frac{\sin x}{\sin x} \cdot \frac{\sin x}{\sin \left(-x\right)}}} \]

   4. Simplified 1.1

      \[\leadsto \color{blue}{\frac{\cos x + -1}{\sin x \cdot \left(\sin x \cdot \frac{\sin x}{\sin x \cdot \sin \left(-x\right)}\right)}} \]
      Proof

      [Start] 0.9	\[ \frac{\frac{\cos x + -1}{\sin x}}{\frac{\sin x}{\sin x} \cdot \frac{\sin x}{\sin \left(-x\right)}} \]
      rational_best-simplify-48 [=>] 0.9	\[ \color{blue}{\frac{\cos x + -1}{\left(\frac{\sin x}{\sin x} \cdot \frac{\sin x}{\sin \left(-x\right)}\right) \cdot \sin x}} \]
      rational_best-simplify-2 [=>] 0.9	\[ \frac{\cos x + -1}{\color{blue}{\sin x \cdot \left(\frac{\sin x}{\sin x} \cdot \frac{\sin x}{\sin \left(-x\right)}\right)}} \]
      rational_best-simplify-61 [=>] 0.9	\[ \frac{\cos x + -1}{\sin x \cdot \color{blue}{\frac{\frac{\sin x}{\sin \left(-x\right)}}{\frac{\sin x}{\sin x}}}} \]
      rational_best-simplify-48 [=>] 0.9	\[ \frac{\cos x + -1}{\sin x \cdot \color{blue}{\frac{\sin x}{\frac{\sin x}{\sin x} \cdot \sin \left(-x\right)}}} \]
      rational_best-simplify-5 [<=] 0.9	\[ \frac{\cos x + -1}{\sin x \cdot \frac{\color{blue}{1 \cdot \sin x}}{\frac{\sin x}{\sin x} \cdot \sin \left(-x\right)}} \]
      rational_best-simplify-49 [=>] 1.0	\[ \frac{\cos x + -1}{\sin x \cdot \color{blue}{\left(\sin x \cdot \frac{1}{\frac{\sin x}{\sin x} \cdot \sin \left(-x\right)}\right)}} \]
      rational_best-simplify-49 [<=] 0.9	\[ \frac{\cos x + -1}{\sin x \cdot \color{blue}{\frac{1 \cdot \sin x}{\frac{\sin x}{\sin x} \cdot \sin \left(-x\right)}}} \]
      rational_best-simplify-5 [=>] 0.9	\[ \frac{\cos x + -1}{\sin x \cdot \frac{\color{blue}{\sin x}}{\frac{\sin x}{\sin x} \cdot \sin \left(-x\right)}} \]
      rational_best-simplify-48 [<=] 0.9	\[ \frac{\cos x + -1}{\sin x \cdot \color{blue}{\frac{\frac{\sin x}{\sin \left(-x\right)}}{\frac{\sin x}{\sin x}}}} \]
      rational_best-simplify-61 [<=] 0.9	\[ \frac{\cos x + -1}{\sin x \cdot \color{blue}{\left(\frac{\sin x}{\sin x} \cdot \frac{\sin x}{\sin \left(-x\right)}\right)}} \]
      rational_best-simplify-41 [=>] 0.9	\[ \frac{\cos x + -1}{\sin x \cdot \color{blue}{\frac{\sin x \cdot \sin x}{\sin \left(-x\right) \cdot \sin x}}} \]
      rational_best-simplify-49 [=>] 1.1	\[ \frac{\cos x + -1}{\sin x \cdot \color{blue}{\left(\sin x \cdot \frac{\sin x}{\sin \left(-x\right) \cdot \sin x}\right)}} \]
      rational_best-simplify-2 [=>] 1.1	\[ \frac{\cos x + -1}{\sin x \cdot \left(\sin x \cdot \frac{\sin x}{\color{blue}{\sin x \cdot \sin \left(-x\right)}}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.02:\\ \;\;\;\;\frac{1 - \cos x}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 0.0005:\\ \;\;\;\;0.5 \cdot x + \left(0.041666666666666664 \cdot {x}^{3} + 0.004166666666666667 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x + -1}{\sin x \cdot \left(\sin x \cdot \frac{\sin x}{\sin x \cdot \sin \left(-x\right)}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	46280

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{\sin x}\\ t_1 := \frac{\cos x}{-2}\\ \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0.0005:\\ \;\;\;\;0.5 \cdot x + \left(0.041666666666666664 \cdot {x}^{3} + 0.004166666666666667 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(t_1 + 1\right)}{\sin x}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	40008

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{\sin x}\\ \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0.0005:\\ \;\;\;\;0.5 \cdot x + \left(0.041666666666666664 \cdot {x}^{3} + 0.004166666666666667 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.7
Cost	39496

\[\begin{array}{l} t_0 := \frac{1 - \cos x}{\sin x}\\ \mathbf{if}\;t_0 \leq -0.02:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0.0005:\\ \;\;\;\;0.5 \cdot x + 0.041666666666666664 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	26.6
Cost	7176

\[\begin{array}{l} t_0 := \frac{1}{\sin x} - \frac{1}{x}\\ \mathbf{if}\;x \leq -2.8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.7:\\ \;\;\;\;0.5 \cdot x + 0.041666666666666664 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	26.7
Cost	7112

\[\begin{array}{l} t_0 := \frac{1}{\sin x} - \frac{1}{x}\\ \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 42:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	61.7
Cost	192

\[-0.16666666666666666 \cdot x \]

Alternative 7
Error	31.8
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023097 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))